40

FARE?

mm a? PHGNQN AND Etamwnmcmw
ENEGQED iémfifiéficfi mmmafis cm was
‘nfimamwea a)? ms
mmsz‘mm mama
mg? a
LA‘E‘TECE mmwcg a? cmmas ws‘m
macaw éM-wEEW CENYERS

Them for the Degree of 931. D.
MICHEGAR S'i‘A'E‘E fiNIVERSI'E‘Y
Hans Rudolf Fankhauser
i969

' MESIS

 

gum a: m‘ 1
I. f .

LISP/11":

Mi" liigan State
University

  
 

1111 1111111111 111173111

319 0096

  

This is to certify that the

thesis entitled

EFFECT OF PHONON AND ELECTRON-ELECTRON INDUCED INTERBAND
TRANSITIONS ON THE THERMOPOWER OF THE TRANSITION METALS

LATTICE DYNAMICS OF CRYSTALS WITH MOLECULAR IMPURITY CENTERS

presented by

Hans Rudolf Fankhauser

has been accepted towards fulfillment
of the requirements for

PhoDo degree in PhZSiCS

flfflm’“

\fMajor professor

Date JUIY 28; 1969

O-169

 

 

ABSTRACT
PART I

EFFECT OF PHONON AND ELECTRON-ELECTRON INDUCED INTERBAND
TRANSITIONS ON THE THERMOPOWER OF THE TRANSITION METALS

PART II

LATTICE DYNAMICS OF CRYSTALS WITH MOLECULAR IMPURITY CENTERS

BY

Hans Rudolf Fankhauser

Part I:

Using a two-band model for the conduction electrons of the transi-
tion metals and assuming that only the lighter carriers contribute to charge

transport the effects of phonon induced and electron-electron interband s-d
transitions are investigated. Provided that the total thermOpower - not

including the phonon-drag contribution - is given by ST = fil-'Z WiSi'we
T i
find that interband electron-electron scattering may manifest itself in the

total thermOpower at low as well as at high temperatures. At lowest tempera-
tures (near T/eD = 0.03), depending upon the magnitudes and temperature
dependences of electron-electron and electron-phonon scattering contribu-
tions, a well defined extremum of the order of luV/OK may appear. At high

temperatures the total thermOpower, weighted as indicated above, may be dom-

inated by electron-electron scattering effects, and in that event, will ex-

hibit a T2 temperature dependence. The effect of the impurities are discus-

sed and the theoretical total thermOpower is compared with available exper-

imental data.

Hans R. Fankhauser

Part II:

The use of symmetry properties results in a great saving of time
and effort in the theoretical study of molucules and crystals and, fre-
quently, the application of group theory leads to valuable qualitative

conclusions. A.group theoretical method to obtain the apprOpriate

eigenvectors of the dynamical problem (normal modes) is presented in

detail and compatibility conditions for the eigenvectors of the sub-

groups are derived in a number of important cases. As a first example of

the practical value of symmetry arguments it is demonstrated that a
study of the dependence of the infrared absorption on polarization

relative to the crystallographic axes already leads to specific infor-
mation on the orientation of a polyatomic molecule imbedded in a cubic

crystal. In a second example we study the scattering of lattice waves

by a stereosc0pic defect molecule. We give a survey on the relevant

aspects of lattice dynamics and show how the molecular coordinates are

removed using the extended Green's function technique. A scattering

formalism is developed and a formally exact solution of the scattering

problem is given in terms of the T matrix. An eXpression for the differ-

ential cross section is derived. It contains two terms, the direct term

and an interference term, which may be of the same order. The reso-

nances in the scattering cross section are given by the resenances in the

T matrix and conditions for such resonances to occur are briefly dis-

cussed. As a simple model a rigid sphere is coupled to a simple cubic

lattice with tangential as well as radial springs. The eigenvalue prob-

lems are solved and the T matrix constructed. The form of the matrix

Hans R. Fankhauser

elements gives information on the possible initial and final states and

on the acoustical activity of the possible modes. For a specific case we

estimate the magnitude of the interference term due to a librational mode
Finally we replace the sphere by
The

and the motion of the center—of-mass.
a rigid ellipsoid which reduces the symmetry at the defect site.
analysis of this case is restricted to librational modes only. We con-

clude with a discussion on what we might expect in a more realistic sit-

uation.

PART I

EFFECT OF PHONON AND ELECTRON-ELECTRON INDUCED INTERBAND
TRANSITIONS ON THE THERMOPOWER OF THE TRANSITION METALS

PART II
LATTICE DYNAMICS OF CRYSTALS WITH MOLECULAR.IMPURITY CENTERS
By

Hans Rudolf Fankhauser

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Physics

1969

Meinem unvergesslichen Vater

ii

ACKNOWLEDGMENTS

I wish to express my sincere gratitude to Professor Frank J. Blatt
for inviting me to complete my studies here at Michigan State University
and for his help and encouragement throughout the course of this work.

The analysis reported in Part I was iniciated by the author, who
focused his attention on electron-electron scattering processes only, at
the ETH, Zurich in 196M. Subsequently he and Dr. L. Colquitt, Jr. collabor-
ated on an extension of this work so as to include phonon induced inter-
band transitions. Many valuable and illuminating diacuSSions with Dr.
Colquitt are gratefully acknowledged.

Concerning the lattice dynamics aspect of Part II I benefitted
to a large extent from many enlightening discussions with Dr. W.M. Hartmann.

For the financial support, I thank the National Science Foundation.

I should also like to thank Mrs. Marie E. Ross for the careful
typing of the manuscript.

Last but not least I Should like to express my deep appreciation

to my dear wife Birgit for all the patience and moral support I got while

I was completing this work.

iii

TABLE OF CONTENTS

Part I:

Section Page
I. Introduction 0 O O O O O O C O O O O O O ' O O O I O O C O 1
II. Phonon Scattering . . . . . . . . . . . . . . . . . . . 5

III. Electron-Electron Scattering . . . . . . . . . . . . . . 10

IV. Total Thermopower . . . . . . . . . . . . . . . . . . . 16
V. Discussion . . C C O - . . . O . . . C . . . O . . O . . 21+
Appendix . O C O O O O C C C C C O C O O C O O O O O 31

References . . . . . . . . . . . . . . . . . . . . . . 35

Part II:

Section
I. IntrodUCt ion . O C O C O C O C C C O C C C C C C . C O C 37
II. Use of group theory to determine the eigenvectors . . . 40

III. Compatibility conditions . . . . . . . . . . . . . . . . 57
IV. Application I - Linear Molecules . . . . . . . . . . . . 66
V. Application II - Stereoscopic Molecules . . . . . . . . 7O
Lattice Dynamics . . . . . . . . . . . . . . . . . . . . 7O
Scattering Formalism . . . . . . . . . . . . . . . . . . 7h
Spherical Molecules . . . . . . . . . . . . . . . . . . 82
Ellipsoidal Molecules . . . . . . . . . . . . . . . . . 100

VI. DiSUSSion O O O O O 0 O O O O O O O O O O O O O O O O 109

References 0 O O O O O O O O O O O O O O O O O O O O O 115

iv

Table

I.

II.

III.

IV.

V.

VI.

VII.

VIII.

IX.

XI.

XII.

TABLES

Stable subspaces of the full cubic group Oh’ structure
A . . . . . . . . . . . . . . . . . . . . . . . . . .
Stable subspaces of the subgroup th’ structure A . .
Stable subSpaces of the subgroup D3d’ structure A. . .
Stable subspaces of the subgroup DZh’ structure A . .
Stable subSpaces of the full cubic group Oh’ structure
B . . . . . . . . . . . . . . . . . . . . . . . . . .
Stable subSpaces of the subgroup Td’ structure B . . .
Stable subspaces of the subgroup D3d’ structure B . .
Stable subspaces of the full cubic group Oh’ structure
C . . . . . . . . . . . . . . . . . . . . . . . . . .
Stable subspaces of the subgroup DZh’ structure C . .

Compatibility conditions for the subgroup DLLh with the
h-fold rotation axis along the z-axis of the cube in
case of structure A. . . . . . . . . . . . . . . . . .
Compatibility conditions for the subgroup D3d with the
3-fold rotation axis along the [lll]-direction of the

cube in case of structure A. . . . . . . . . . .

Compatibility conditions for the subgroup DZh’ with the

2-fold C-axis oriented along the [llO]-direction of the

cube in case of structure A. . .

Page

M6
#7
MB

l+9
so
51

52

53
55

61

62

Table

XIII.

XIV.

TABLES (continued)

Page
Compatibility conditions for the stable subspaces which
correspond to the representation according to which the
infrared active modes transform in case of structure B
and symmetry D3d' The 3-fold rotation axis is oriented
along the [lll]-direction of the cube . . . . . . . . . 63
Compatibility conditions for the stable subspaces which
correspond to the representation according to which the
infrared active modes transform in case of structure C
and symmetry DZh' The 2-fold C-axis is oriented along

the [llO]-direction of the cube. . . . . . . . . . . . . 6h

vi

Part I:

Figure

Figure

Figure

Figure

Figure

Figure

1.

2.

5.

FIGURES

Page
Theoretical total thermopower in the case of an invert-
ed d-band, ks<kd, md/mS = 10, for different gap par-
ameter n. O C O O O O O O O O O O O O O O O O O O O O 18

Theoretical total thermopower in the case of an invert-

ed d-band, ks<kd, md/ms = 10, n = 0.1. In the inset,

the details of a local maximum - not to be confused

with a phonon-drag peak - are shown. This extremum is
associated with the exponential decay of phonon induced

s-d transitions at very low temperatures. . . . . . . 19
Theoretical total thermopower in the case of an invert-

ed d-band, ks>kd, md/mS = 10, n = 0.5. In the inset,

the details of a local minimum, due to electron-electron
scattering effects, are shown. . . . . . . . . . . . 21
Theoretical total thermopower in the case where the two
bands have the same curvature, ks<kd, n = 0.1, for

Ind/ms = 3, 10. o o o o o o o o o o o o o o o o o o o 22

Experimental thermopower of some of the transition metals
taken from N. Cusack and P. Kendall, Ref. 21. . . . . . 25
Theoretical total thermopower in the case where the two
bands have the same curvature kg>kd, md/mS = 3, n = 0.5,

for different values of the ratio R. . . . . . . . . . 29

vii

Part II:
Figure la).
Figure lb).

Figure 1c).

Figure 2.

Figure 3.

FIGURES (continued)
Page

simple cubic structure (A).
body-centered cubic structure (B).
face-centered cubic structure (C).
The heavy bonds indicate possible orientations
of a linear molecule with the center-of-mass at
the origin . . . . . . . . . . . . . . . . . . . . . . hl
ll -<b2 e g(0)l-2, which enters the scattering
cross section of an isotope defect, as a function
of the lattice wave frequency in the Debye approx?
imation, taken from Thoma and Ludwig [37]. . . . . . . 97
It (F18)|2’ which enters the scattering cross
section of the librational mode, as a function
of the square modulus of the lattice wave

frequency in the Debye approximation, taken from

Wagner [25]. . . . . . . . . . . . . . . . . . . . . . 99

viii

PART I

EFFECT OF PHONON AND ELECTRON-ELECTRON INDUCED INTERBAND
TRANSITIONS ON THE THERMOPOWER OF THE TRANSITION METALS

I. INTRODUCTION
Recently there has been a resurgence of interest in the low temp-

erature resistivities of transition metalslIh. Although it has been known

5

for some time that the electrical resistivity, p, of some of the transi-
tion metals varies as T2 at the lowest temperatures, concomitant linear
variations in the thermal resistivities, W, have only recently been obser-
ved. The origin of the T2 dependence of p is a problem of léng standing.
Although it was in 1937 that Baber6 proposed that electron-electron scat-
tering was the cause for this variation, evidence to establish this pro-
posal as valid has been slow in coming. Two of the major criticism of
Baber's proposal are:
(a) p is observed to vary as T2 in only a few of the
transition metals rather than all of them as might
be expected.
(b) Experimentally it is found that electron-electron-

scattering contributes a T2 term to p and consequent-

ly dominates the total resistivity at lowest temper-

atures where the contribution from phonon-scattering

eventually varies as T5. Similarly at the highest

temperatures where the lattice resistivity varies line-

arly with temperature the electron-electron scatter-

ing contribution should again be dominant. This

latter behavior, however, has not been observed.

In a study on these problems Colquitt7

attempted to answer the
first objection by a careful analysis of available experimental data. He

1

was able to show that one could give a consistent theoretical interpre—
tation of the ideal resistivities of the transition metals in terms of a
two band model and assuming (in all of the metals) the existence of a T2
term which may, however, be masked to a greater or lesser degree by phonon
scattering in different metals of the series.

Appel8 attempted to answer the second objection to electron-
electron scattering by appealing to numerical estimates of the two re-
sistivity contributions. He argues that in some metals the T2 contri-
bution will only "peak-through" the phonon contribution at extremely high
temperatures - near or above the melting point.

With increasing evidence for and interest in e-e scattering in
transition metals, it seemed appropriate to consider the effect of these
scattering processes on another electron transport phenomenon, the thermo-
electric power. The calculation has been carried forward within the frame-
work of the two-band model introduced by Mott9 many years ago. Although
the work assumes the "standard band structure" for the two bands, we have
extended Mott's model somewhat by placing no a priori restriction on the
curvatures of the bands; i.e., either the "s-band" or the "d-band" may be
electron-like or hole-like. We alsoallow’kszkd as well as Es<hd where kg,
k6 are the Fermi wave vectors. Hence we consider four different situ-
ations, corresponding to two bands of identical or oppositecurvature,
with Egakd and ks<kd. The subscripts s and d in this paper are used simp
ply to denote a light mass, conduction band and a high mass, narrow band,
respectively.

As we shall see, it is not possible to classify the results unique—

ly in terms of the above-mentioned four possibilities since two other

 

3

parameters, the effective mass ratio md/m8 and the "momentum gap" n defined
by Eq. (6) which was introduced by Calquitt7 have a profound effect on the
results.

In most situations electron-phonon scattering dominates over elec-
tron-electron interband scattering in its effect on the thermopower at all
temperatures and the temperature dependence of the thermopower is linear
at elevated temperatures (T/eD > 1). However, when the momentum gap is not
too small, say 0.3 or more, we do find instances where the electron-elec-
tron contribution to the thermopower exhibits a well-defined extremum at
very low temperatures. We also find conditions under which electron-
electron interband scattering may dominate the effect of phonon-induced
scattering at high temperatures and manifest itself in a more rapid temp-

I erature dependence (roughly T2) of the total thermopower.

In this investigation phonon-drag was completely neglected. A
more severe limitation, however, is the neglect of Umklapp processes
which in electron-electron scattering, at any rate, do not occur frequent-
ly enough to modify the transport coefficients significantlylo. The
reason appears to be that energy conservation severely restricts the pos-
sibility of electron-electron Umklapp processes, in contrast to phonon-
phonon or phonon-electron Umklapp processes. However, since Umklapp pro-
cesses depend sensitively on the details of the Fermi surface, it seemed
to us that to include these processes in the parabolic band approximation
would still not answer the difficult question of their importance in a
more realistic situation. The calculation is, thus, in the spirit of a
model calculation and we concern ourselves only with general qualitative

conclusions.

 

In sections 11 and III the effects of electron-phonon and electron-
electron scattering on the different intrinsic transport properties are stud-
ied. In section IV the temperature dependence and sign of the total thermo-
power are discussed and figures for some typical cases are shown. In
section V the effect of electron-electron scattering on the total thermo-
power at low and at elevated temperature is discussed and the results are

compared with available experimental data.

 

II. PHONON SCATTERING

The effects of electron-phonon scattering on the electrical and

thermal resistivities of the transition metals in terms of a two band model

are given by11’7.

3

. P h _ -2/ 2

ppho(T) = 8 SB 172 2 2 (g )3 2 1/3“
81r(2m8) eEF D D

9
+ “’d EFF—q [J3(-i:) “IX-$1

and

27m P Sh3'1‘
W s s

3
ph ho(T W16n5( )I7ZEE (9D) (k: T)2

e e e
D -1/3 -2/3 T 2 2 D _ D
[J5(—T) + 2 n (—913) [2/3 a J5(-—T) 1/3 J7(--T)]

m P 9 9
d sd D E
+ (Dd Egg; 2/3 [J5 (—f) -J5 (—T) 1

9 9
+ 2/3 3:2 [J3(—]T)) “J3(—:)]}

Here n is the effective number of the lighter carriers per atom,

statistical weight (degeneracy) of the d-states, Ps

(1)

(2)

is the

s and Psd are proportion-

al to the square of the matrix elements for phonon-induced 3-3 and 87d tran-

sitions respectively, E is the Fermi energy, 6

F

kBeE the minimum energy of phonons that can induce s-d transitions. The

transport integrals Jn(x) are defined in Eq. (7).

D the Debye temperature, and

In an early work Mott9 argues that the resistivities (electrical

and thermal) due to phonon induced s-d transitions would contain a factor

5

 

 

 

6

Nd(EF), the density of states in the d—band. Wilson11 on the other hand,
showed that if not all states on the d-Fermi sphere could be reached from
a given s-state by phonon induced transitions, the proportionality factor
should be¢mdmd. In Mott's case, one assumes that the upper limit of the
phonon wave vector, LEI, inducing s-d transitions is equal to Es + Ed,
ks, Ed being the Fermi momenta of s- and d-type carriers reapectively.
In the other case, the upper limit is the Debye wave vector, I3D"

There is little distinction between these two cases when one
is computing the magnitudes of the resistivities. However, as pointed
out by Wilson, the thermopowers in the two cases are very different. In
the first case in which one assumes that the largest momentum transferred
is Siax = Es + Ed < an, a situation which seems hardly realized in nature,
the thermopower would be augmented by a factor proportioned to 5Nd(€)/B€
which always has the same sign. In the second case where the largest
momentum transferred is Siix =‘SD <‘Es + Ed’ the thermopower will contain
a contribution from BBB/Be which may be positive or negative depending on
the relative magnitudes of the Fermi momenta and the relative curvatures
of the two bands (See Eq. (5)).

We shall restrict ourselves to the latter case so that Wilson's
model is the appropriate one. This is the reason that in the Eqs. (1)
and (2) 6D appears in the transport integrals instead of hu(ks + Ed)/kB,
where g.is the velocity of sound.
We now assume that the following expression, derived by Zimanlz

for low temperatures is valid also when we allow interband as well as

intraband transitions

 

 

 

 

2 2
Spho = 3e ' L as + (1 ' L 8e ’ 2 \ T) ( ) L (3)
o 0 2x EF SD 0 €=E

F

We note that in the high temperature limit Eq. (3) reduces to the well-

known formula

 

22
fl kBT Bln
Spho = ' 3eEF EF 5e e=EF (h)

We now substitute the electrical resistivity as given by Eq. (1) into
Eq. (3). We assume, of course, that the two scattering processes, intra-

band and interband, are independent and contribute additively to the total

 

 

 

 

 

resistivity. For the Fermi surface, 0, and the Fermi momentum, EF’ we put
in the corresponding values of the lighter carriers and obtain
2
n 2k T L
S h z B 2E + E1 2;
p 0 3e F o F
T 2
(5—) d -1 1
D
A 9J5(—)+ Ind----2-(—D-)[(-'11)-;l:|:—k +k](/T)
+ msEF SDhZ s
T
(5-) d
D D E
AmE J,_(T) m [J3( ) J3(T) .
F s
L 9 2 k 2
- _§ _3__ (.12 ('2 (5)
Lo anE T
F
neing the relation
9E his-Edi
THE" = $1 (6)
D D

 

 

 

: rt FE!

Alt...

8

to calculate the derivative er/Be. The following quantities and abbrevia-

tions have been introduced

 

 

 

Jn(x) = k/p Zn dz
° (ez-1)(1-e'z)
(61:2)2/3 r12 PSS
A = hag P (7)
“’d sd
(GE/T)3
C(eE/T) = (BE/T) -(6E/T)
[e -1][1 - e 1

where a is the lattice parameter and Lo’ LT’ Ls’ are the Lorentz numbers

 

 

defined by

2 k
n B 2

Lo — 3 (e )

L _ pss + pad (8)

T Tsts + wsd)

LS = pss

T WSS

The upper sign in the numerator in Eq. (5) corresponds to the case Ed<§s’

the lower one to kd>ks, j is equal to 2 in the case of an inverted d-band,
and equal to 1 otherwise.

The sign of e is that of the lighter carriers. In our model we
assume only that the carriers described by one band are substantially

heavier than those of the other and that the former do not contribute to the

charge transport. The dominant charge carriers may be either electrons or

holes.

 

 

9

The discussion of the effect on the total thermopower of phonon
induced scattering is complicated by the fact, that the intrinsic thermo-
power must be weighted by the corresponding thermal resistivity. What
we expect is that for T/6fi>1 the curly bracket in Eq. (5) will be constant

and S therefore proportional to T. Below this temperature the contri-

pho

bution decreases mainly because the ratio of the Lorentz numbers diminishes.

Below 6E the exponential decay of s-d transitions further reduces the con-

tribution and Spho may even reverse sign.

 

 

III. ELECTRON-ELECTRON SCATTERING
If we assume that one can define a relaxation time for these
processes, then the change with time of the distribution of the carriers

due to collisions is given by13,

 

 

o
Bflv .3) fear) - f (2.3;)
t coll — 7(221) (9)
If we denote by11+
51' s'
2
.15 9.‘ 5L' (10)
the a priori transition probability that an electron in state k1 collides

with an electron in state (k2,,kz + gkz ) and that the two particles are
scattered into the states (kq,% +dkp ), (k2,fl k' + dk' 2) respectively,
and furthermore assume the electrons to be free, and describe the inter-

1
action by a screened Coulomb potential, then we obtain 5

 

 

r}, i = _ 32 1:3 en W k1+k2’ k3+kh2 dAZ dA3 dAh
1 coll hk BTVZV 2v3_,+ N [Ik3- -;c,1l2+322 ]
MG<€1+€23f61h)fonO-(l f3 °)(1-fh°) dc;2 de3 den (11)

Here the subscripts 1, 2, 3, and h, stand for‘kl,‘k2,‘k&, and 5%, respec-

tively. The vi's are the corresponding Fermi velocities, V is the volume
I

of the Brillouin zone, ab stands for ($1 + $2 ' $3 ‘ ¢h) where the $1

are defined by

10

 

 

O Bf:
f = -
1 fi ¢i 561 (12)
g, is the reciprocal of the screening radius, 5k the Kronecker delta and

the surface elements dAi are defined in the Appendix. For a discussion
of the properties of the energy conservation function 0(6) we refer to
Ziman16.

We should like to mention that implicit in Eq. (11) is the fact
that as a result of momentum conservation, Normal intraband transitions
provide no relaxation.

For the calculational details, we refer the reader to the

Appendix and quote here merely the result

 

f 128 useh
£1 11 = - ‘hhk TVZV v v
C° B —2—3—l+

[(nkBT)2+(el-EF)2] fo(el)[ {-1 f °(e) ”ls—TE? 1(Am’in Amax) (13)

where B = l/kBT and the integral 1(Aminiamax) is given by Eqs. (Alh) and

(A15).
We now use Eqs. (9), (12), (13) and the property
0 O O
afi = - w (11+)
52: kBT
to obtain
“7V2 1 IR I 1 (15)

 

 

1031) = T1112 T3111” TA min’ Amax) (nkBT)2+(el-EF)2

 

12

In the usual framework of the theory of macroscopic transport coefficients,

the electrical conductivity is given to first order by

 

0(a) -.- e2 fv r(k ) dA (16)
1219,. -1. “1 1

Inserting Eq. (15) in Eq. (16) we obtain

 

 

. 4
F 38hn9e2 m1m2m3mh (k T)L I(‘f‘l’min’a‘mafl

B

From this expression we see that the most effective scattering processes
are those in which (s,d)-—>(d',d"). The contribution of this type of
process is larger than those of any other electron~electron scattering
processes by a factor greater than Nd(EF)/NS(EF)° Hereafter we restrict

our attention to these processes only and obtain for the electrical resis-

 

tivity
38hn9e2 3 1(Ahin’amax) . , 2 , 8
pe-e = 2 msmd 9 )kBI) )1 )
h V 3%
The lower and upper limits of I(Amin’Amax) depend on the relative magni-
tudes of k and k as follows
-s -d
1is < Ed : Amin L lid - Es A‘max = Ed + 1("s (19a)
. = . m 13 a 2k 1 b
Es > lid ° A'min Es Ed max ‘-d ( 9 )

We require, of course, that Amin < Amax and hence in case of Es >‘Ed we

 

13
get the condition

.3 < 3rd

If this condition is violated, then there is no way for a scattering process

to occur conserving linear momentum. Substituting the appropriate values

 

for 1(A‘in% x) according to Eq. (A15) we finally obtain
forks < Ed
— A m “‘3 (39.1) (l-x) 1 -1 2
pe-e _ ewe ; 2 - 2 +‘i tan (2X?) T (203)
1535‘, (x+1) + x2 (l-x) + 112
and for k > Ed

at 111

MHZ + 1 (l A
where X = llk = kS/kd and we have set 8.: kg. The constant factor is

Z 2
= 192n9e kB (21)

e-e --7;-—'-'
h V2

From Eq. (h) we now find the intrinsic thermopower due to electron-electron

scattering

for is < Ed

 

 

 

 

1h

 

 

 

 

 

 

 

 

 

 

 

 

 

S _ nzkgT md
8'6
3 e 2 2
*1 a
g m ‘_ m m
“If-(x1.1)2] (f X’1 + 1) [x2-(1-X)2](-m—S x1 + 1) 2 52 161(2x2+1)+ux‘
d r d " d "
+
J [x2 + 091)ij [x2 + (1-30212 1 + ux"
. Xklz g - -———l%zL—-+ X71 tan.1 2X2
1 (X+1) +X“ (1-30 .08 J
2
- k mS/md if (233)
and for Es >‘Ed
S = _ nzkgT md
e-e 3 e fisz
.8
'1 2 2 m8 -1 ms 2 -l ‘
(:21 (111-1) [1-(1-1) “3-, 1 ) E— (1.411 )+67~1-5>~ -11
d d
(Axel-ma [1+(1-x)2]2 [1+2).(1->.)124-(3x-1)2 h m ,md (22b)
2). _ 1-1. + tan-1 x-1 ‘ S
. 11121.1 (1-1)2+1 “2" 1") .

 

 

In Eqs. (22a, b) the upper sign corresponds to the case of an inverted

d-band.
The total measured thermopower (discounting phonon-drag) is the

sum of the intrinsic thermopowers each weighted by the corresponding ther-

17

mal resistivity. For electron-electron scattering we have

15

W = per (23)
k 2

T(12-n2)(_§)
e
Since the expressions for Se-e and we-e are rather complicated,
it is difficult to predict the magnitude and sign of this contribution to
the total thermopower in the general case. We do expect that if this con-
tribution dominates that associated with electron-phonon scattering at high

temperatures, the total thermopower will vary as QT + BT2, where the second

 

term arises from electron-electron scattering. This follows from the ex- r

pression for the total thermopower1

Spho + We-e Sewe (2h)

+
wpho We-e

S = wpho

D). th0 at high temperatures is

independent of T whereas We e and Se—e are both linear in T. At still

and the fact that th0 (9D) >> We_e(9

higher temperatures We-e may become comparable to, or greater than, tho
and where this happens, the quadratic contribution in the total thermo-
power will diminish. In that event, the total thermopower will exhibit a
linear temperature dependence even though electron-electron scattering
effects dominate over those of electron-phonon scattering. In some cases,
this behavior is apparent from the calculated results and also in the data

in some of the transition metals (See Sections IV and V).

IV. TOTAL THERMOPOWER

The total thermopower for multiple scattering mechanisms is given

by18
1
ST = fi-' 2 W181
T 1
-with (25)
W = Z W
T 1 1

where Si and W1 are the contributions to the thermopower and thermal
resistivities of each mechanism independently. Thus, before we can
construct the total thermopower, it is necessary to know the relative
magnitudes of We-e and.tho. As it is difficult to estimate these from
first principles, we have resorted to an empirical estimate of the ratio

Ppho(T)/pe_e(T) by defining a parameter TE by

ppho(TE) = Pe-e(TE) (26)

5

Estimates from experimental data put T in the range from 5°K to 20°K

E
consistent with the evidence that we-e(9D)<wpho( 6D). The theoretical
total thermopower is plotted in Figures 1, 2, 3, and h, representing
typical cases for different values of the gap parameter, n, different
effective mass ratios, md/ms, and possible arrangements of the s- and
d-bands.
T T .
To aid our discussion we introduce Spho and Se-e’ the weight-

ed contributions of the two scattering processes to the total thermo-

Power. These are defined by

16

 

17

S
_ _EEE__EEE ST _ We-e Se-e
W e-e — WT

We now consider four distinct situations

Case I: One band inverted relative to the other band.

19

ST is negative at all temperatures and

pho

it is dominant at low temperatures. For

small and intermediate values of n (50.5)

ST is positive. When m /m is large
e-e d s

at higher temper-

T
e-e

(10) it dominates Sgho
atures. In the case of large n (0.7) S
is positive for md/ms = 10 only, and it is
always smaller than Sgho throughout the
temperature range. These results are
shown in Figure 1. If n = 0.1 and md/mS

= 10 we find a local extremum of the total
thermopower at very low temperatures asso-
ciated with the freezing out of phonon in-

duced s-d transitions. A typical curve is

shown in Figure 2. 0n the other hand, in

the case of n = 0.3 and the same large ratio

md/m we find (depending upon the magnitude
3

of k ) a local extremum due to electron-
-s

electron scattering processes. Both these

(27)

 

l8

1‘

.c nous-=31 new usououuuo a on .3 ~60.
o... cl 0 nouoofi. . an
_ ouuo>ca so no some as» ea uoaooosuucu Houou as u
a o a so .. o .
nouao~|s\a.quoao

B-

 

mt:

 

£11m

19

.ouuauouoeauu sea auo> us noouuuonouu

on. oousosu sosose mo havoc unausoaoaxo on» sous venouoouno nu asaouuxo uqsfi .ssogu on.

1 goon monocooawsa a

.u.o .. r .2 u an...

111...

caused on o» no: 1 Bonuses aooou a mo uuuuuoo any .uuou:« can an
aosoouo oouuo>uu so we some any ad nosoeoauoau Houou duoauouoosh

 

.N Pagan rnl

 

wt.» .. a m 1. ..-

1| d“ ”m% 1(r.
n:

9.0

.ng

.0.

 

(>3. im

20

peculiarities will be discussed more in
detail below (See section V.).

Sgho is always negative and it is dominant at
low temperatures. SZ-e is always negative
and dominates Sgho at higher temperatures.
Independent of the ratio md/ms we find for

n = 0.5 a strong local extremum associated
with electron-electron scattering effects.

A representative curve for this behaviour

is shown in Figure 3.

Case II: Both bands have curvatures of equal sign.

a)

k
-s

< k

-d .

T
pho

the temperature range. For small and inter-

S as well as 8:_e are negative throughout

mediate values of n and for large md/m8

ST dominates at higher temperatures, but

e-e
this is not the case if md/ms is small (3).
This change of the temperature dependence
of the total thermopower with the effective
mass ratio is shown in Figure h.

T l t eratures The
8 ho is dominant at ow emp .

P
sign is negative20 if md/m8 is small and
n is small or intermediate (5 0.3), or if

md/mS is large and n is small (0.1). The

sign is positive if md/m8 is small and n

.muommwo mcfiuomum

.90 .1. c .2 u

s\oe

.o

lml
xAx

w-

.mBosm mum

on couuoofio1couuomao cu moo «EDEHCHE HmooH m mo mHHmumo ecu .uummcw ecu CH

Aocmnuo wouuo>cw no mo ammo ecu a“ umBoeoEumcu Hmuou HmofiumuomcH

 

.m muswfim

 

22

0) K)
(H u

m .11qu
H E\wE new «H.o H r anxV x

I}

.

impoum>uoo ofimm ecu m>ms mocmc csu use mumsa mmmu ecu CH umxoooEuuzu Meson HmofluouomsH .t muswam

OthBQE:

n unE\vE

b

i.

y-

.

wOVI

1O”!

.owu

 

£1 *1 km

is large (2’0.5), or if md/ms is large and
n is intermediate or large (2_0.3). Indepen-

dent of the effective mass ratio s:_e is

positive for small and intermediate n (S 0.5)

T
pho

Under the same conditions as in case Ia) we

and dominates S at high temperatures. P
obtain a local extremum characteristic for
the exponential decay of the phonon induced
s-d transitions at very low temperatures.
We also find local extrema due to electron-
electron scattering effects which become
more pronounced as the gap size increases

(TIZ 0.3) and the ratio md/mS becomes lar-

ger.

V. DISCUSSION
There are several important limitations to our calculations which
preclude a detailed comparison with the experimental data for each of the
transition metals. First, we have used a spherical model for the Fermi
surfaces of the conduction electrons in order to simplify the calculations.

Although this is an obvious oversimplification of the actual Fermi sur-

 

faces in the transition metals, it perhaps suffices to represent the
general features of these metals. The magnitudes of the quantities md/ms,

a, 3D, ks, Ed, n and related derivatives with respect to the energy which

 

enter the theory must then, however, be considered as empirical parameters.
Secondly, we have omitted considerations of phonon-drag processes. Con-
sequently, a comparison with the experimental data must be restricted to
regions where T/eD is greater or much less than unity and phonon-drag
effects have essentially disappeared. Finally, we neglected Umklapp
processes throughout this investigation.

Nonetheless, there are certain general features of the experi-
mental data in these two limiting regions which seem to bear out our
model calculation. For comparison we include the figure given by Cusack
and Kendallz1 (Figure 5) and refer also to more recent resultszz. In
the high temperature limit the thermopower for the transition metals is
observed23’21 to vary from large negative values (e.g. for Pd and Pt)
to large positive values (e.g. for W and Mo) at a given temperature as
we pass from one metal to another. Although the argument that this
variation is due to differences in the slope of the density of states of

the d-band is essentially correct (i.e. making no distinction between‘

the Mott and Wilson models), it may be crucial in some cases to include

1
r‘.
!'- LL

 

 

.HN .emm .Haaesmx .m

com xommoo .2 Beam coxou mHmuoE cowuwmcmuu can wo QEOm mo uoaonoEumcu Hmucmsauomxm .m ouswflm

 

oov.
n

 

 

 

the effects of electron-electron scattering. For example, the thermo-

power of W and Mo above the Debye temperature is given quite closely by

sT = aT - BT2 (28)

'2 uV/OK, e a»2 . 10'"5 uv/(°K)2,

where the constants are on a 11.5 ° 10
respectively. The second term may reflect the importance of electron-
electron scattering on the thermOpower at elevated temperatures. More-
over, in the low temperature region (near 10°K) the experimental data
for W22 display a peak of the order of 0.2 uV/oK which may be due to
effects of electronaelectron interband scattering. To understand this,
we must look at the weighted contribution to the total thermopower,
since from the linear temperature dependence of the corresponding in-

trinsic thermopower one would not eXpect such a behavior. From Eq° (27)

we get the following temperature dependence

2
T AT
Se_e = -;f--- (29)
BT + CT

since we know that Se-e’ as well as we-e’ are proportional to the tem-
perature. In the case where we have intraband scattering induced by
phonons only (e.g. noble metals), n would be equal to 2. In our case
where in the temperature region of interest the probability of phonon
induced interband transitions drOps eXponentially, n will be larger than
2 but to a first approximation (up to the second term in the expansion
of the exponential factor) still smaller than 3. We now differentiate
with respect to temperature and obtain the following relation for the

temperature at which the weighted contribution of electron-electron

 

 

effects reaches a local extremum

Textr = [ n~2,B (30)

The calculated thermoPower exhibits such a local extremum only if

Textr lies below the characteristic temperature where effects due to

phonon induced interband scattering are diminished exponentially. Other-

 

wise S? . not only diminishes with increasing temperature but the extremum
{rm-J

1"

will further be masked by ST
pho

temperature. In the case of ks >‘Ed the local extremum becomes more

pronounced as the ratio K = Ed/Es approaches 1/3 for then only large

which increases rapidly with increasing

 

angle scattering events provide relaxation.

If the "momentum gap" is small (n = 0.1), the phonon induced
s-d transitions decay exponentially only at very low temperatures after
essentially all contributions from electroneelectron scattering effects
have diminished considerably. It then is not surprising,considering the

complex temperature dependence of ST

pho in this region,that we may find

under these circumstances and especially for a large ratio md/mS a local
extremum quite similar to the one ascribed to electronwelectron scattering
effects above.

Thus a local extremum in the case of small n is more likely
to be associated with phonon induced scattering effects whereas in case
of intermediate or large n it might be due to the influence of electron-
electron scattering.

We also might point out that in view of the rather complicated

temperature dependence of ST

pho’ especially at low temperatures, and the

28

interplay with S:_e we must not be surprised if the general behaviour of
the total thermopower in this region is such that the thermopower, though
it must surely vanish at absolute zero, does not appear to extrapolate

to this value even if measurements are carried out to quite low temper-
atures, e.g. near 1°K2h.

We also should like to mention briefly the influence of effects
due to impurities. Since the corresponding thermal resistivity is pro-
portional to Tm1 and the intrinsic thermopower varies linearly with the
temperature, we expect no qualitative change at higher temperature, but
only a parallel shift in very impure materials. 0n the other hand, at
low temperature there may arise a substantial change especially in the
case where we have a local extremum in the ideal case. This situation

is indicated in Figure 6 where we show 8 versus T for various values of

P(293°K)/pres = R

We should like to point out that Figures luh and 6 were
obtained with an almost random choice (within our assumptions) of the
parameters involved. In various portions of the temperature scale they
qualitatively reflect some of the features of the experimental data
shown in Figure 5. A better fit to the experimental results could be
obtained by adjusting the lattice constant, a(3°10-10m), the statistical
weight of Sad transitions in the case of phonon.induced scattering,

O -l
m

)

deSd/PSS (2), the magnitudevof the Fermi vector ks (O.L7 to 1.88°101

and the Debye wave vector, 3D(1-6°1010mu1)o

The values in parenthesis
indicate our choice and were not changed with temperature.
In view of the various simplifying assumptions of the model

calculation, such adjustment of parameters is of questionable value.

 

 

 

 

\) unskm
.m oaumu man we mvsam> unmumMMHc now «u.c u r «n n t\ e

u. . . «a . ouswa
auMme ouaum>u=o osmm osu mbm: means can ecu muons ammo ecu :H umaomoenmsu Hmuoo Hmofiuo ooze m .m_

&x—- W. v. ”-

 

|xFLalIl ;
P b

 

o_umnl.ll.
ngamllll
non ..... -.i
09!“ .

 

\ rm

 

30

What we do wish to emphasize is that this simple model is capable of re-

1

producing the general features of the available experimental data.

 

 

a? P

E

ND

..L

X

 

 

 

 

APPENDIX
We start with Eq. (11)
. h k
f z: n. 3 fl +0.12, ~ +~
‘51 - h 3‘; 9 ¢ “.127“ 1 3 2“ dAZdA3dAh
coll f1 kBTV XZ’X'V 1+ [3'15 klltgj

[[OKCI +6 -C 3.61) flo/ f: o-(l f<3 0--)(1 f1!» o)de dC3dC.

and perform first the integration over energy. e made use of the

relation

.0 .o e(Ci-EF)B
1 1

and of the property of 0(a) for large times to reduce the energy depend~

ent part to the form

(CQQEF>B

' dr
1 JF d€3 Jr 7 e I 3 ‘
r1-EF;s (CQ-EFjfi (CszFJB \ .rl+cgecq-EF)B

1+e 6 J +1 (e +1) (e “ +1?

 

 

 

To evaluate the integral over CO we make the substitutions

L:

(c -e )8 (en-E )B
e 3 z a and e a F = u

Separation into partial fraction

 

 

 

l _ 1 l _ 1 ]
(u+1)(au+l) ’ l-a u+l u+ é
gives m
-e
1 1n u+l e1 §
-:\ 1 ' e e B
(la’e “+3 :1 3 -1
0

(A1)

"\
21>
M

v

(A3)

(A5)

(A6)

 

32

With the further substitutions

(€lnEF)B
e = b and (cl-€3)B = x (A7)

we find for the whole energy dependent part of Eq. (A1)25

2 3
(nkBT) + (El-EF)

 

 

 

b 2 0
“7' [fl‘ + (1n b)“] e
2 g (e -E )B '(E 'E )5
(b+1) B 2(e 1 F +1) (e 1 F +1)
rag [(nkBT)2 + (cl-EF)2] f°<c1) [1-f°<c1)1 (A8)

 

This leaves us with the momentum dependent part of Eq. (A1). We now

define
P 2 2 Z n
AI — lls3-kll - s1 + .53 - c 12153 cpl
(A9)
.2 _ _ 2 _ 2 2 - . y
‘11 Lhz'bhl “ 52 + 5’ 2 5254 COS q’Ii

where the Q's are the angles between the corresponding k~vectors. First,

 

we keep k1 and k2 fixed and vary kh only; then
dA = ZnK 2 sin m dm = 2: ”II 2 . (A10)
14- "4+ II II E.
E kn A I

where K1 is the radius of the Fermi sphere of carriers of type i.

Similarly 2

dA = K 2 dag = :§—~ dA(k2wkm) (All)

2 "2

La;
La

where [2.2 isthe Space angle generated by 152 (while 151+ varies) and Aflsz-ku)
is the surface area swept out by the rotation of‘kzzku. Thus, the integra-

tions dA2, dAh can be evaluated as shown below

 

 

 

 

 

 

 

2 2
dA
2n U/:---3—--\jp fEEEE—- giL- 5 dAI
2 2 - k ’ -
[was as. A; A1 An I was)
dA3
= 211: £251» f 2 2 (A12)
AIM. +51
In an analogous manner we write
2 . A1 2
dA3= 21:.I_<__3 sm (pl er1— 2:: E153 53 dAI (A13)
and-find that Eq. (A12) becomes
A.
M. 2 £212.39. e1
2“ $251+ 2 2 2 = h“ k 2 2..
AI[AI +3] -1 A [AI +5]
min
K K K
= u -—3-—-E1 . I (Amin’Amax) (A )
The integral I (Amin’ Amax) so defined is given byc
.A
max
1(A1,A)=J§[2A2+étan-1§J (A15)
m n max
min

Combining the above results we obtain finally

3h

 

 

. E 1;»
128n/e 2 H
‘k . = - ‘ . [<nk I) + (c -E.)&1 f°<c >[1-£°(e )1
“1 coll hAk TVder,V B 1 F l 1
B -.:-3-LL
§,K Kh

$5 fif— I (Amin, Amax) (A16)

We may point out here if we had not excluded intraband scat-
tering already, their contribution to f would now be seen to vanish

kl coll
since in that case, particles 1, 2, and 3, A are indistinguishable and

consequently AI,II and dAI, must vanish (m1 : QII = n in Eqs. (A9) or

II
(A10), (A13))-

 

 

 

 

LIST OF REFERENCES

‘0 S” T“ 9‘

10.
11.

12.

13.

1h.

15.
16.

17.

18.

19.

REFERENCES

G. K. White and R. J. Tainsh, Phys. Rev. Letters 1 , 165 (1967).
C. Herring, Phys. Rev. Letters 1:5 167, 68% (E) (1967).
J. T. Schriempf, Phys. Rev. Letters 12, 1131 (1967).

A. C. Anderson, R. E. Peterson, and J. E. Robichaux, Phys. Rev.
Letters 29, M59 (1968).

G. K. White and S. B. Woods, Phil. Trans. Roy. Soc. (London) A251,
273 (1958).

w. G. Baber, Proc. Roy. Soc. (London) gigg, 383 (1937).
L. Colquitt, Jr., J. Appl. Phys. 3E5 2u5u (1965).

J. Appel, Phil. Mag. 8, 1071 (1963).

N. F. Mott, Proc. Roy. Soc. (London) A126, 368 (1936).
J. Appel, Phys. Rev. 123: 1760 (1961).

A. H. Wilson, Proc. Roy. Soc. (London) élél: 580 (1938).

J. Ziman, Electrons and Phonons (Oxford University Press, Oxford

1963) PP- 1103-

e.g. A. H. Wilson, Theory of Metals (Cambridge University Press,
Cambridge 1953) 2nd ed. p.-6.

J. Ziman, Ref. 12, pp. 257.
J. Ziman, Ref. 12, pp. h12.

J. Ziman, Ref. 12, p. 129.

F. J. Blatt and H. R. Fankhauser, Phys. kondens. Materie jg 183
(1965).

D. K. C. McDonald, Thermoelectricity_(J. Wiley and.Sons, New York
1962) p. 107.

In a few exceptional situations Sgho may be slightly positive at
extremely low temperatures.

Except at very low temperatures where it may undergo one or two
sign changes.

U.)
U]

 

 

21+.

25.

26.

REFERENCES (continued)
N. Cusack and P. Kendall, Proc. Phys. Soc. (London) 72, 898 (1958).

R. L. Carter, A. I. Davidson and P. A. Schroeder, to be published in
J. Phys. Chem. Solids.

G. Borelius, Handbuch der Metallphysik (Ed. by Prof. Dr. G. Masing,
Akademische Verlagsgesellschaft, Leipzig 1935) p. 185.

H. J. Trodahl, Rev. Sci. Instrum. fig, 6M8 (1969). [see particularly
Figure 7].

D. Bierens de Haan, Nouvelles Tables d'Integrales Définies (Hafner
Publishing Company, New York 1939) Ed. of 1867 - corrected, p. 1L8.

 

H. B. Dwight, Tables of Inte r31; and other Mathematical Data (The
MacMillan Company, New York 19 5) 3th ed. p. 30.

 

36

 

 

 

 

PART I I

LATTICE DYNAMICS OF CRYSTALS WITH MOLECULAR IMPURITY CENTERS

I. INTRODUCTION

During the past decade there has developed considerable interest
in the study of vibrational spectra of imperfect crystals [1]. The reason
for pursuing these investigations is two-fold. First, the effect of the
impurity is generally to introduce localized or resonance (pseudo localized)
modes in the vibrational spectrum of the ideal lattice. These frequently
give rise to observable changes in bulk properties, for example, specific
heat [2], resistivity [3], and infrared absorption [1], and a detailed
study of these modes can provide useful information on interatomic forces
between the impurity and host lattice ions. Second, if the impurity has
internal degrees of freedom, the impurity - host lattice interaction can
affect a change of the normal modes associated with these degrees of
freedom. This is a matter of considerable practical importance since one
method often employed in infrared and Raman spectroscopy is to introduce
the molecule of interest in a suitable matrix, usually an alkali halide
crystal. Depending upon the strength of the interaction between this
molecule and the surrounding matrix and the orientation of the defect
molecule with respect to the crystallographic axes the recorded spectra

will not be characteristic of the free molecule [h-8].

It is generally recognized that a great saving in time and effort
can be achieved in the theoretical study of these systems by making optimum

use of symmetry properties. Not only does the application of group theory

expedite detailed calculations but it also frequently leads to valuable

qualitative deduction based on symmetry considerations alone. We'shall

37

 

 

38

demonstrate in section II the group theoretical procedure employed in the
solution of such problems in detail, giving not only the decomposition into
the irreducible representations of the apprOpriate subgroup [9] but also the
corresponding basis vectors in explicit form. Using the symmetry elements
themselves to obtain the stable subspaces rather than a projection Operator

technique has the advantage that one does not need an explicit matrix repre-

 

sentation of the symmetry operations and furthermore only a few symmetry
elements are needed for the complete reduction of the total space. The
stable subspaces as well as instructive compatibility conditions (section
III) for a number of important cases are given in tabular form. As a first

example, we shall demonstrate in section IV that a study of the dependence

 

of the infrared absorption on polarization relative to the crystallographic
axes already leads to specific information on the orientation of a poly-
atomic molecule imbedded in a cubic crystal. In a second example, in
section V, we shall make optimal use of the symmetry properties while
studying the scattering of lattice waves by a stereoscopic defect molecule.
In a first subsection on lattice dynamics we give a survey of Wagner‘s
treatment [2%, 25] which is most suitable to solve this type of problems.
The molecular coordinates are removed by means of a Green's function tech-
nique and we are left with a problem of the same dimension as in case of a
point defect. However, the difference is that in our case the effective
disturbance is complimented by a term which has poles at the molecular
frequencies. In the next subsection we develop a scattering formalism and
give a formally exact solution of the scattering problem in terms of the T

matrix. An expression for the differential cross section is derived. It

39

contains two terms, the direct term and an interference term, which may be
of equal importance. From the form of the two terms it is seen that the
scattering processes of phonons are far more complicated than the scatter-
ing of plane waves by a static potential. Conditions for such resonances
to occur are briefly discussed.

We then consider the following simple model. A.rigid sphere is

 

coupled to a simple cubic lattice with tangential as well as radial springs.
The eigenvalue problems are solved using the stable subspaces in section II.
With this information we construct for each mode the scattering matrix and

calculate the matrix elements to obtain the scattering cross section. From

 

the form of these matrix elements we can determine possible initial and final
states and decide if the mode is acoustically active. We discuss the con-
ditions under which there may be inband modes but focus our attention to

the modes transforming according to the irreduciblerepresentationsFlg
(librational motion) and Flu (motion of the center-of-mass) and estimate
the magnitude of the interference term for a specific case. In the next
subsection we replace the sphere by a rigid ellipsoid with one moment of
inertia different from the other two. In this case the symmetry at the de-
fect site is reduced depending upon the orientation of the molecule. We
restrict our analysis to librational modes only. We then conclude with a

discussion of some of the details and what we expect in a more realistic

situation.

II. USE OF GROUP THEORY TO DETERMINE THE EIGENVECTORS

We shall consider the following three basic structures given in
Fig. 1.a), b), c). The dimension of the space carrying the total represen-
tation ST is given by the number of points involved. This space is generh‘
ated by all allowed point group operations as well as translations. For
a molecule imbedded in a crystal we must not exclude free rotation opera-
tions since these yield the librational modes. The translation of the
center-of-mass must be removed, but this is most conveniently done by ex-
cluding that set of eigenvectors from the total space which correspond to
this motion at the end of the analysis. This results in lowering the dimen-
sion by three of the reducible subspace which carries that (those) irreduc-
ible representation(s) for which the coordinate axes transform according
to the three degrees of freedom of the center-of-mass. With the aid of
character tables [9, 10] we decompose the total representation of the
symmetry group (or subgroup) G into its irreducible representations and

determine their multiplicities mu from Frobenius' theorem

1 *
mu ='§ g G X(X) X u(X) (1)

e
where g is the number of elements in G (order of G) and X9(x) is the
character of the nth irreducible representation. From group theoretical
theorems [9, 10, 11] we conclude that if the symmetry group involved has

c classes then the carrier space of the total representation will decom-
pose into at least c subspaces, the carrier spaces of the row(s) of the

c irreducible representations. The dimension du of the th irreducible
representation gives the degree of its degeneracy due to symmetry, 1. e.,

the number of subspaces with the same eigenvalue. The multiplicity m;

ho

 

 

 

 

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of the nth irreducible representation in the total representation gives the
dimension of the corresponding subSpaces which is also the dimension of the
eigenvalue problem which we have to solve. These considerations lead, in
a natural way, to the correct dynamical eigenvectors which are linear com-

binations of the vectors which Span the stable subspace carrying a particu-

 

lar row of a certain irreducible representation. The idea goes back to
some general remarks by Wigner [12] and was also used by Ludwig [l3] and 1
Dettmann and Ludwig [1h]. The point is that one uses the symmetry ele- {H
ments of the group to decompose the total space ST into its stable con- h
stituents by separating out the subspaces which are spanned by the set of

all eigenvectors corresponding either to the eigenvalue +1 or -1 under a

specific symmetry operation. The intersection of two spaces obtained in

this way is also an invariant subspace. In this manner one gets subspaces,

characterized by the eigenvalues of the symmetry operations, by success-

ively operating with commuting group elements and forming intersections

until the subspace has the correct dimension. Symmetry elements not used

in this procedure can not lead to a further reduction because the multi-

plicity is determined with consideration of the full symmetry group. This

is the reason (and advantage compared to projector technique where one has

to have an explicit matrix representation of all the symmetry elements)

that one needs only a few of the symmetry elements.

In order to introduce our notation let us consider the operation

of the inversion I on the total space corresponding to the structure A

ST . {3;};3;I2¥2}2ng;T&lz|OxOyOzllxlylzlzxzyzzl3x3y3z} (2)
1 s; .-{3x3ygzlzx2yzzl1x1ylzloxoyozl1x1ylz 2x2y22|3x3y3z} (3)

Where the curly bracket is a compact notation for the set of the vectors

1+3

spanning the 21-dimensiona1 space. The numbers label the lattice points and
x, y, z are the components of the displacements from equilibrium position.

In what follows it is important always to bear in mind that whatever stands

in the 21322'32, say, as introduced in the curly bracket of Eq. (2) represents
the displacement of the point 3 in the z-direction, hence ....'aablOOb}
means that the displacements of the lattice point 2 in the x and y direction
are related and equal a, the displacement of point 2 in the z direction is

b and is opposite to the z displacement of point 3 which is the only possible

motion for the latter point. From Eq. (3) we determine by inspection the

 

two subspaces invariant under inversion, one corresponding to the eigen-

value +1, one to -1, respectively:

A _._._. _._... _._ _
Sf x3y3z3 xzyzzzlxlylzllo O O (xlylzllxzyzzzlx3y323 (ha)
i = {X3Y3z3lxzyzzzlxlylzllxoyoxolxlylzllxzyZ221x3Y3z3} (Ab)

and we note that in case of inversion symmetry the displacements at a point

n g? = (11K ny n2) and that of the inverted point 5 53': (3% E& E?) are re-

lated by
n H
E =' 6
n ‘3
Eu 2' Sn (5b)

for the eigenvalues +1 (g = gerade) and -1 (u = ungerade) respectively.
In order to demonstrate how one obtains the intersection of two stable
subspaces we first determine the stable subspace of another symmetry
element, oz, say, a reflection in the mirror plane perpendicular to the

z axis

uh

Oz 3:» = {3x3y-3zlzxzy-2 1 1 -1 0x0 x y

z x y 2 ~02 1x1y-1z zxzy-zzl3 3 ~32} (6)

Y

with the stable subspaces

83;: {x3y3zélx5y50IxhyuO'xoyoOlxlylolxzyZle3y323} (7a)
Sé {£372 '0 Oz (0 Oz '0 OZolO Oz '0 Oz [xy 2; (7b)
o-z 3 3 3 5 h 1 2 3 3 3
for the eigenvalues +1 and -1, respectively. The intersection
$162: sgn ‘83: is then given by the parts of the subspaces which are
compatible with one-another, i.e., that subspace which is stable
under inversion, either with eigenvalue +1 or -1, as well as under the
operation 02 corresponding to a certain eigenvalue. Let us concentrate
on the following two cases
S’f‘gz = Sg‘n sg-‘z = {0 0 Z3|§2§20 l§1§10l0 0 0 'xlyIO (xzyzlo 0 23} (8a)

A A
s-+ -
» BIA 8%

and suppose that we are dealing with full cubic (octahedreal) symmetry.

Then from the character table for the group Oh we see that the carrier

spaces S“ for the irreducible representations A18, A28, and E3 respective-
A

1y are subSpaces of the intersection Sf; , i.e., S’ 13( Sf; etc.,
2 2

whereas the intersection S-+ contains the stable subspaces carrying

IO
2
the irreducible representations F1u and FZu respectively, i.e.,
Flu F2u
S C 8'3 and S C S-s- . The dimensions of the intersections obtained

I I
z z

 

 

1+5

here are in this case higher than the corresponding multiplicities of the
irreducible representations and one has to proceed with other symmetry

elements in a similar fashion.

In tables I to IX the stable subspaces, which are not normal-
ized, are listed for a number of important cases. The ones for the full
cubic group Oh are given for all three structures shown in Fig. 1. For
the structure of type A (Fig. 1a) the decomposition of the 21 dimensional
total space into its stable subspaces is presented for the subgroups
th’ D3d and DZh' As examples of structure B the subgroups Td and D3d
are considered. Finally, the 39 dimensional carrier space of the total
representation corresponding to the structure C is split into its stable
constituents for the subgroup D2h'

In the first column only those irreducible representations
of the (sub)group are listed, which are part of the total representation.
In the next three columns the multiplicities m: of the corresponding ir-
reducible representations are given. The first of these is for the case
where one allows for vibrations only, the next corresponds to librational

(quasi-rotational) motion only; and the last includes all degrees of free-

dom. Clearly, the difference m:

m: = m; - (m: + m:) (9)

is associated with the translational degrees of freedom and we now have
to exclude translations of the center-of-mass explicitly.

As an example let us consider the 3 dimensional stable sub-
spate carrying the first row of the irreducible representation F1u of

Oh (table 1.). From the information given in the table we see that we

 

 

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50

 

 

 

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51

 

 

 

 

 

 

 

 

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52

 

 

 

 

 

 

 

 

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IT
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53

 

 

 

 

 

 

 

ANN H mmvw N Na
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55

ha

Cl

56

have to reduce the dimension of this carrier Space by one. If the set of
points involved is occupied by atoms of equal mass then the obvious, neces-
sary but not sufficient, condition is that the following equality not be

satisfied

x = x I x, o (10)

The necessary and sufficient condition is

5: e“ = 0, not all 5" = 0 (11)

N

i.e., that the components of the displacements be linearly dependent. This

condition is in the special caSe which we are considering

x0 + 2x1 + hxz = O . (12)

However, the reduction of the dimension of the carrier space in question
follows in a more natural way if one looks at the dynamical problem. As
mentioned above the multiplicity m? of the uth irreducible representation
in the total representation gives the dimension of the eigenvalue problem
which we have to solve. One of the three roots of the secular equation in
our example will be zero and it is that set of eigenvectors, which is asso-
ciated with this particular eigenvalue, which we have to exclude.

The remaining columns give the components of the displacements
which span the m” dimensional stable subSpace associated with the nth

T

irreducible representation. Frequent use of the relations (5a, b) was made.

III. COMPATIBILITY CONDITIONS

There are two obvious physical situations for which compatibility
conditions similar to the compatibility relations derived in band theory
of crystals [15] will be useful. First, if one introduces a polyatomic
molecule into a crystal lattice then generally the symmetry of the system
is reduced. One then wishes to determine first if any of the degenerate
representations are split, and secondly, what restrictions are imposed
on the corresponding eigenvectors under the new circumstances. The same
questions also arise if one applies stress, an external electric or a
magnetic field to a crystal. The first part of the problem is answered
by the correlation tables for the species of a group and its subgroups
given in the literature [9], whereas the answer to the latter part is
more difficult since the eigenvectors found in the literature are usually
presented in pictorial form [16]. The problem is then to establish a
set of linear equations relating the components of the stable sub-
spaces of the full cubic group with those of the carrier spaces of the
irreducible representation of the subgroup into which the irreducible
representations of the group of higher order decompose according to the
correlation table. This is done using the stable subspaces given in
section II. This procedure often leads to a reduction of the free
parameters of the stable subspaces involved since the stable subspaces
of a subgroup frequently have a higher dimension than the corresponding
stable subSpaces of the group of higher order. As an example, let us
consider in the case of structure B the conditions imposed on the modes

transforming according to the irreducible representations A2u or Eu

57

58 ..

into which a mode transforming according to the first row of the irreducible

representation F1u splits if the symmetry of the system is lowered from Oh

 

l
to D3d' From tables V and VII we use the corresponding stable subspaces.
A2u ‘ {ya y2 ley2 x2 yzlxz yz y2|x1 x1 x1"“ x° x°l (13‘)
E1 ° {x' x'2i'lx' x' z'lx' x' z'lx' x'2§"]x' x'2§"I (13b)
u°1+1Iu222222111°°°
E2 . {in xII 0 IX” 'i'II Ell'xfl “in 2",?" xII O [in x" O l (13C)
11 ° 2 2 2 2 1 1 ° °
1 ._ _. ._ ._
Flu ' {*1 Y1 y1'"1 y1 y1'"1 y'1 y1'x1 3'1 3'1"“ O O I (11*)

We note that the stable subspace carrying the first row of the irreducible

representation F1 is 3 dimensional whereas the stable subspaces carrying

lu

A2u and Eu, respectively, are k and 5 dimensional, respectively. This
means that the relations between the components of the stable subspaces
corresponding to the lower symmetry may not contain more than 3 free
parameters, a, b, c, respectively. These relations and restrictions are

what we call the compatibility conditions. The sets of the relevant linear

equations are

x0 + x5 - x3 = a (15a)

x0 + x3 + x3 = 0 (15b)

xo -2x$ = 0 (15¢)

x1+xi'x'1'=b (16a)
I_ ll:

x1 + x1 x1 c (16b)

X “ZXi = C (16C)

59

I II __
x2 + x2 + x2 _ b (173)
y2 + xé - x; = -c (17b)
y2 + zé + z; = -c (17C)
YE + Xé + x5 = b (17d)
x2 + xé - x5 = -c (l7e)
,2. .5 = . (m)
Y2 + Xi ‘ X" = b (18a)
y2 + xi + x" = c (18b)
x2 - 2xfi = -c (18c)
with the solutions
1 1 N 1
x0 = 3a , X0 = 63 ,9 X0 =--2-a (19 a” b) C)
x = l(b+2c) x' = 1(b-c) x" ='l(c- b) (20 a b c)
1 3 ’ 1 6 ’ 1 2 ’ ’
x Nib x'=-1-(b-3c) x; —-]-‘-(b+c) (21. b c)
2 3 ’ 2 6 ’ 2 ’ ’
1 1 I,
y2 — 8P , zé =~§ , 22 =-c (22 a, b, c)
x' -Il(b+3c) x" =‘l(c-b) . (23 a b)
L 6 ’ 2 ’

In tables X to XIV we list these conditions in full for the subgroups th’ D3d

and DZh in case of a structure of type A; for the subgroup D * with structure B,

3d
as well as for the subgroup D2h with structure C we will give the compati-
bility conditions only for those irreducible representations according to
Which the infrared active modes transform.

In the first column the irreducible representations of the full

cubic group are listed and in the next column the correlation table of the

..‘lI' 6

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63

 

NIHmnH NomIHIN KSIIHNu N «NNNHINN NoaH INN

 

 

 

 

 

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6S

respective subgroup is reproduced. For all components of the stable sub-
spaces of the subgroup the relations imposed by the group of higher symmetry
are tabulated under the appropriate heading. A.bar (-) means that the par-
ticular stable subspace is not contained in the subspace carrying a certain
row of the irreducible representation given at the left even though it is
contained in the union of the stable subspaces carrying the different rows
of the same irreducible representation in agreement with the correlation
table.

Since the stable subspaces are given in explicit form in
section III it should not be difficult for the reader to establish the
missing compatibility relations for the cubic group or derive them for

the case when the group of highest symmetry is not the full cubic group.

IV. APPLICATION I -_LINEAR MOLECULES

We now apply the results to the case of a linear triatomic mole-
cule in a cubic crystal. If the molecule is alined along one of the cubic
axes, the z-axis, say, then the appropriate symmetry group (subgroup of
Oh) is th and the applicable structure is of type A. On the other hand,
if the molecule is oriented along a body diagonal, the [lll]-direction,
the symmetry of the system is reduced to D3d and the associated structure
is of type B. The third case which we shall consider is the molecule
oriented parallel to a face diagonal which leads to the symmetry D2h and
belongs to structure C. Let us concentrate on the infrared active modes

2: and “u (in the group D they transform according to the irreducible

dh

representations A and Eu’ respectively) of this molecule. In case

Zu
of full cubic symmetry (Oh) the infrared active modes transform accord-
ing to one of the rows of the 3 dimensional irreducible representation
Flu’ and, therefore are 3-fold degenerate. Clearly, if we introduce
this molecule into a cubic crystal then its infrared active modes have
to have the same transformation properties and hence form a base for

the irreducible representation F This feature makes it unnecessary

In.
to derive compatibility relations especially for Duh and its subgroups
th’ D3d and D2h respectively, and we can use the ones derived above
(tables.x, XIII and XIV, respectively).

In the first case the total space is spanned by
{3x 3y 32'Ox Oy 02'3x 3y 3;}. Imposing the compatibility conditions

given in table x on the stable subspaces3listed in table II we are

_ left with the following

66

67

DLLh : s 2“ = {OObIOOaIOOb} , a + 2b

E

s u = {boolaoolboo} , a + 2b
E2

3 u = {ObOIOaOIObo} , a + 2b

0 : stretching in z-direction (2ha)

H

ll

0 : bending in (OlO)-plane (Zhb)

o : bending in (100)-p1ane (eke)

Similarly for the other two cases where the

total Space is spanned by
{1"I ‘I lo 0 o [1 1 1 } we find
x y z x y z x y z

A
D3d : S 2“ = {bbblaaafbbb} , a + 2b

0 : stretching in

[lll]-direction (25a)
1

Eu —

s = {bbzblaazalbbzb} , a + 2b = o : bending in (110)-plane (25b)
2
U

s = {bb olaa olbb o} , a + 2b = o : bending in (1fI)-p1ane (25c)

DZh : s 1“ = {bb olaa olbb o} ,2bI+ a

= O : stretching in
[110] direction. (26a)
8B2u = {bb Olaa Olbb 0} ,2b + a = O : bending in (OOl)-plane (26b)
Bsu .
s = {o Oblo salo 0b} ,2b + a = o

: bending in (llO)-plane (26c)

The condition a + 2b = 0 represents the exclusion of the translation of the

center-of-mass. In the first two cases the degenerate mode (nu) does not

split whereas in the third case the appropriate symmetry group has one

dimensional (non-degenerate) representations only, and, therefore, the

previously degenerate mode must split. We are not surprised to find in

each case a stretching mode with the same orientation as the molecule and
this fact may, in many cases, be sufficient to determine the orientation
of the molecule. Experimentally we would detect this by an absorption

maximum if the exciting radiation is polarized parallel to the orientation

of the molecule. We must remember, however, that group theory can only

68

supply the necessary condition that a particular mode be infrared active,
and the above mentioned stretching mode need not be active, or if it is,
could be so weak to be experimentally unobservable. Then, in our ideali-
zed case, we can still extract enough information from the bending modes
to uniquely determine the orientation of the molecule. The simplest case
to detect, of course, is the splitting of the degenerate bending mode if
the molecule is oriented along the face diagonal of the cube.

The other two possibilities are easily resolved by the fol-
lowing experiment. We use light polarized linearly in a plane perpendi-
cular to one of the cubic axes. If the molecule happens to be oriented
along this particular axis then the absorption is a maximum and inde-
pendent of a rotation of the system about this axis. If we repeat the
same experiment along one of the two other h-fold axes we should find
a sinusoidal dependence of the absorption upon rotation.

If the molecule is oriented along the [llll-direction a
rotation of the polarization vector in the (lOO)-plane would again
give a uniform absorption. However, if one rotates the polarization
vector in the (IEO)-plane the minimum will appear in this case when the
polarization vector is along [111] as contrasted with the previous
situation when the minimum appeared along the [OOll-direction.

Clearly, the above arguments for the ideal case, where we
assumed that gll_the molecules have the figmg_orientation, does not apply
to a real situation where the molecules will be distributed at random
among all possible equivalent orientations. This random distribution

has the effect that we observe an average absorption for any orientation

69

of the polarization vector, even for the stretching modes. Consequently,
we cannot, in the realistic case, distinguish between molecules oriented
randomly along <lll> and <IOO> directions,respective1y. Of course, the
splitting of the bending modes for the <llO> orientation does provide

a means for establishing this orientation.

V. APPLICATION II - STEREOSCOPIC MOLECULES

Lattice Dynamics

The Green's function formalism introduced by Lifshitz [l7] and
others [18 - 23] for the calculation of lattice vibrations in impure crys-
tals is restricted to disturbed lattices with an unchanged number of parti-
cles (monatomic impurity\centens), i.e., to cases where there are neither
new degrees of freedom nor a change in symmetry at this particular lattice
site. Wagner [2%, 25] extended this method to molecular impurity centers
and in this subsection we shall give a survey of this work.

If a molecule of s+l masses m replaces a regular lattice atom at
n = O, we may transform the molecular variables to a new set (xi, gi, ... 5:)
where x: gives the position of the center-of-mass of the molecule. The e:
may be chosen rather arbitrarily, but they must diagonalize the kinetic ener-
gy, with an associated effective mass m:. The three center-of—mass coordi-
nates, xi, and the total mass of the molecule, Mo = m1 + m2 ... ms+1, are
added to the other lattice coordinates x: and masses M: (M: = ideal masses),
establishing a 3N-dimensional system as in the ideal case. Then, there is
a natural way of looking at the problem:

(a) The "lattice system" is characterized by a 3N x 3N matrix

H ij = Ho 11 + H1 ij, where H0 ij describes the unperturbed
nm nm nm nm
lattice.

(b) The "molecular system" is characterized by a 33 x 3s matrix

h ij.
V“
(c) The interaction between the two systems is defined by a

iJ
3N x 33 matrix an .

7O

71

(d) The disturbance is assumed to extend only to a small number

r of lattice sites n around the origin, which implies that ‘

Himij and anijare essentially zero outside this region.

We introduce the substitutions

zi (Mo)l/2 xi
n n n

C: (m:)1/2 5:

.. _ 9
L 13 _ (Mo Mo) l/..
n m
(27)
ij * * -1/2 h ij
aVH (mV mil) W1

ij _ o * -1/2 R ij
an (Mn mv) nv

M

i' 2 2 n o 0 -1/2 1 ij
Am 3(III ) = II. (1 - ;3) 5m son a“. + (Mn Mm) um
n

With this, the eigenvalue equations of the two connected systems are

0 (3N equations) (28a)

(L + A(a)2) - w21)'z + B'C

0 (3s equations) (28b)

A
Q
I
8
H
v
J“:
+
m
N
II

Without the perturbation A.and the coupling B each of the two systems de-

fines a Green's function I
11090 n (M)

(D2 = _(DZI - = (298)
G( ) (L ) E53205” - (02

 

7(w2) = (a - cn'B‘I)’1 — gm“)— (29b)

where n(kk), §(x) denote the normalized eigenvectors,ImZ(§x),(”2(K) the
eigenfrequencies of the matrices L and 05 respectively. The solution

for the ideal lattice

L'n(1.s>~) = New) new (30)

we also shall use in the form
-1 i i -
nine) =N ”saw e“ (31)

where‘k_is the wave vector and k the polarization of the phonon. It is
easily verified that the total Green's function for the combined system

(28a) - (28b) is given by the individual Green's functions Eqs. (29a,b)

in the simple way:
2
GT<w2) = (6'w ) O 2) (32)
0 7(w )
It is not necessary, however, to use this (3N + 38) x (3N + 33) matrix,
because the special structure of the system (28a) - (28b) allows the
molecular coordinates Q to be excluded. Using Eq. (29b) we can write

Eq. (28b) in the form,
2 Iv.
C=-7(a>)Bz (33)
Introducing this expression for C in Eq. (28a) and multiplying from the

left by Gfibz), Eq. (28a) takes the fonm,

2

z = - an?) [A(N2) - a 7<N > if] .2 (3‘4)

and the molecular coordinates are thus removed. Now A and B can be written

as

A =(g g , B = (b, o) (35)

where a is 3r x 3r and b a 3r x 33 matrix, both extending only over the r
lattice sites involved around the molecular defect. Hence, the eigenvalue

equation, extracted from the system (3h), reads:

D<N2> = new + em?) [a - b mg) ‘61) = o (36)

which is a determinant of rank 3r. gflmz) is the 3r x 3r matrix of 6032)
which belongs to the r involved lattice points. It is seen that the

2 . 2 2 '~
Lifshitz matrix I + g0» ) a 18 supplemented by the matrix -gflm ) b 703 ) b,

written more explicitly

( 2) b ( 2)‘S g b b l,n,m = 1,2..3r ( )
8<D 7cm = 8 7 37
1vu n1 1V VH um v,u 1,2..3s

In this formulation the eigenvalue determinant is of the same rank 3r as in
the Lifshitz problem and -b yfibz)‘b may be considered as an additional distur-

bance in the "lattice system". The effective disturbance,

v = a(m2) - b m2) ’5 (38)

within the system of lattice coordinates contains, apart from the rather

smooth function aomz) (associated with the motion of the center-of-mass of

the molecular defect), the additional molecular term - b 70mg) 3 which has

poles at the molecular frequencies<p(x). Thus v cannot be treated as a per-

turbation near the¢m(x) frequencies, however small the coupling b may be.
The 3N + 38 roots of Eq. (36) are the eigensolutions of the

fundamental equations of motion (28a,b). Considering for the moment the

71+

factor gfimz) only, we see from the definition (29a) of the lattice Green's
function that gsz) jumps from -w to +m between two consecutive<m2(kk)
values. This means that there must be a solution of Eq. (36) between two
adjacent<m(kx) values. Thus we have a spectrum of solutions in the same
region and with the same density as in the ideal case. Eventually there
are one or a few solutions outside the ideal band(s); these are the local-
ized modes. The factor sang) is of little influence as it is not a strong-
ly varying function Of<b2. There are some new features if we take into
account the factor b 7032) b. This term has additional poles at the mmle-
cular frequencies<nz(x), and gives rise to 33 new solutions distributed
outside and inside the ideal band(s). But the more important fact is that
the Lifshitz solutions, associated with the matrix ghmz) afinz), are strong-
ly disturbed in the neighborhood of those molecular frequenciesImz(K)

lying inside the band(s). This has a great influence on phonon scatter-

ing and yields resonances in the scattering amplitude.

The Scattering Formalism

The theory of the scattering of lattice waves was first worked
out rigorously by Lifshitz [26]. This theory has subsequently been devel-
oped further by Klein [27,28], Takeno [29], Krumhansl [30] and Callaway
[31].

We have indicated in the last subsection that the spectrum of
solutions for the disturbed lattice occupies the same regions as in the
ideal lattice, apart from the singular solutions outside the band(s) which
we shall not consider here. As the distribution is very dense, we ask for

solutions of the form

75

zyisx) = gen + wgex) <39)

the frequency of which lies very near toIm(kX). In this case niLkX) repre-
sents the incident phonon and w;(kk) the scattered wave. The asymptotic

expression for the latter is
flex, em (no)

which defines the scattering amplitude fi(kk,‘5'k'). As in the quantum
mechanical theory of scattering the differential scattering cross section

and the scattering amplitude are related by
cm, 15') = lies», emlz (1+1)

where the bar reminds us that later on we shall have to sum over the pos-
sible polarizations K' of the final states. However, the proof of an optic-
al theorem does not carry over directly to phonon scattering [28], because
when changes in.mass are involved, there is also a change in the "effective
metric tensor".

The Spectrum of both the ideal and the disturbed solutions is
discrete but very dense. It is convenient, therefore, to go to the continu-

um by replacing summations in k space by integrations

f...d3k (1+2)

-1 9....3
N £...=(2fl) z:

‘kk K
where Va = a3 is the volume of the primitive unit cell, a the lattice spac-
ing. Then the Green's function GGmZ) is no longer defined. In order to

avoid the improper integral we redefine Gflmz) according to standard

76

scattering theory as

0+(N2) = (L - I (N2 + ie))'1

3m) flex) (1&3)

= Z

kk 2 2
~ (1) (15k) - (w+ 16)
which is the Green's function for the "outgoing" wave solution as long as
the group velocity at the stationary points is an outward normal to the
frequency surface. For a detailed discussion on'this matter we refer to

Maradudin [32] and Ludwig [13].

Substituting Eq. (39) and using Eq. (30) we obtain

me) = - G+(w2) we?) n09») - are?) m2) now (M)

where V denotes the effective disturbance v in the total Space. We can

solve this equation by iteration

w+=-G+Vn+G+VG+Vn-G+VG+VG+Vniu (1&5)

If we step with the first term on the right side of this expansion we have
a solution which is equivalent to the first Born approximation of ordinary
scattering theory. The succeeding terms represent the second, third ....,
Born approximations to the scattered wave. We can write a formally exact

solution to Eq. (MS) in terms'of the so-called association or scattering

matrix T
me) = - and?) we?) n(k>~) (1+6)

where T is the solution of the equations

T=V-VGT=V-TGV (1+7)

77

This is a convenient form to calculate the scattered wave, because the
rank of the matrix T is equal to the number of degrees of freedom in the

crystal affected by the introduction of the defect. If we partition T in

T = (t ‘12) (1‘8)

t21 t22

the following manner

where t is defined in the defect Space (space of v), and where, according to

Eq. (#6), t21 = t 2, then substitution of Eq. (#8) into Eq. (#7) yields the

1

result that the matrices t and t are null matrices, and that the

12’ t21 22

matrix t satisfies the following equation in the defect space

n
II

v - vgt = v - tgv

 

(49)
= v(I + gv)-1
which is a 3r x 3r matrix.
According to Eqs. (#3) and (M6) the scattered wave can be
written as
'x‘ t >01 , ,
W(}s>\)=-Z :5 H132 new (50)
+ Elk! a) (btxi)_(w ,_ 16) .

where in the scalar product we have introduced the short notation lgx> to
label a plane wave state n(kx).

' Because of the low rank 3r, it is in general very easy to di-
agonalize the denominator of the t matrix (Eq. (#9)) by symmetry consid-

erations. Let us assume that we know the eigensolutions of the matrix

g+v:

78
g+v-e(v) = u(v) e(v) (51)

where e(v) is a column vector and v labels the row of the irreducible
representation according to which the eigenvectors transform. Thus we can

write the t matrix in the form

N

VeV
+uv ea

e
t _ v g 1

Very often the appropriate symmetry group of the defect is a proper or im-

prOper subgroup of the symmetry group of the host lattice, and in this case

v has the same eigenvectors as g+v:

v-e(v) = v(v) e(v) (53)

and the t matrix can be brought to the form

e :3. 145%?) em em = ’6 cm rm (51+)

where

t(V) = 1 Z 3 V . (55)

and the matrices T(v) are given by the outer product of e(v), €(v), respec-

tively.

Let us now turn back to the scattering problem. We start from a
relation between the scattering.matrix.and the scattering amplitude which
was derived by Ludwig [13] from an asymptotic expression for the scattered

wave. If we make the acoustic approximation
musk) = CO») '5' (56)

then in our notation the scattering amplitude has the form

79

= ia3 1
Mn C2(K')

 

I'm. .Is'x') <,ls'>~' Itlis» .. (57)

As indicated below Eq. (Ml) we need to sum over all branches of the final

states

aw Itlis» (58)

The resulting differential scattering cross section is

0(th ls')= lRisbis'x'HZ =2} lam, L'NHZ

£6
:_1 —"“—§ i, i" i3 TL“ eng'k')ei(k"X")Q)IItlk'>\'>
16x c (K')

d’SIIxII I t IE» (59)

We now use the orthogonality relation

2 e"f(k'>c) e.(,1g'>.") = 5
i 1" 1

xix"

to simplify Eq. (59) and, at the same time, decompose the scattering matrix

according to Eq. (56) with the result

6

a

16x

 

OOSMS') = Z 8 1L— t*(u)t(V) <5'K'IT(H)|}9>
ll v

c (X')

PLM

2

°<}s'?~'IT(v) L15» (61)

Using the fact that the matrix elements in Eq. (61) are real we put it in

final form

 

A, , = a 1 2 , , 2 +
005 Is) 1613 i. mg; |t(u)| <i<, x mole»
2“; Re (awn) <l<,'>~'|T(u)|,ls7~><ls'>~'|T(v)|B%>] (62)

From this expression we see clearly that the resonances of the scattering
cross section are given by the resonances in the t matrix. Furthermore,
we realize that the scattering of lattice waves by an impurity is much
more complicated than the scattering of plane waves by a static potential

2, which determines the stationary

in quantum theory. The equation 052(k)») = a)
points, can have solutions in several branches of the functionIm2(§X). This
has the consequence that although the incoming wave is in a definite branch
of a?(§k), there can be several scattered waves propagating in different
directions with the same frequency but with different group velocities and
polarizations.

In the rest of this subsection we review briefly a discussion by
Klein [27] and also Wagner [25]. The expressions (52) or (59) show that

there is a resonance in the scattering amplitude if the real part of one

of the denominators l + u(v) becomes zero. Hence, the resonance condition
is
l + Re u(v)Qo2) = O (63)

and the resonance frequency we shall denote bbev. If this resonance is

sufficiently strong, then the vth term may exceed all other terms in the

neighborhood ofIn =¢m and we can approximate the matrix t by expanding
V

81

the denominator around a) = wv

t : v(wz) 8“) ‘é'(v)

 

 

for a) z a) (61+)
’
V ((1)2- (1)2)R + 11
V V V
with
RV = J3 Re II(II)(N2)
db cn ab
V
and (65)
2
= I
Iv m u(”(0%)
AS the denominator of Eq. (6h) enters with its absolute square into the
first term for the differential cross section (Eq. (62)), the half-width
of the resonance in this term is given by
w2(; _w2 ‘ I
2 V = 2 V (66)
R
wv (DV V

and there is a Sharp resonance if this expression is much smaller than
unity.

In his analysis Wagner [25] ignored the possibility for the second

term (interference term) in Eq (62) to occur. Even though it is not likely

that a resonance in that term would be as pronounced as one in the first
term, there exists still the potentiality that the two terms might be of

equal importance since the cross term does not enter through a perturba-
tion calculation. If two.modes, which transform according to rows of

different irreducible representations, have eigenfrequencies in the same

82

range than they might contribute appreciably to the scattering cross section.
This possibility exists for instance in the combination of an inband libration-
al mode and the motion of the center-of—mass of the molecular defect.

From the Special form of the disturbance (Eq. (38)) we can see
that some of the u(v)'s (at least one) must contain the poles of the molecu-
lar Green's function 7. Since these vary over a wide range, they are very
likely to give a solution of the resonance condition Eq. (63). On the other
hand, there may be some of the v(v)'s which do not contain the molecular
poles for which there also exist a solution of the resonance condition.

To calculate the structure and Spectral position of the resonances
explicitly, we have to establish a specific model for both the lattice and
the molecular defect.

What we do expect, however, is that the shape and magnitude of
the resonances have the same dependence on the density of the frequency
Spectrum of the host lattice at the position where these pseudolocalized
modes would like to appear as in the case of a point defect. The analysis
of Dawber and Elliott [33] shows that the resonances due to a monatomic
impurity is more pronounced the lower the density of the frequency dis-

tribution of the ideal lattice at this particular frequency.

Spherical Molecules

We know that the internal binding in a molecule is often much
stronger than the binding to the host lattice and it is practically un-
changed when the molecule is brought into the lattice.

If we assume such strong internal binding, we can distinguish

83

three types of motion for the molecular defect:

(a)

(b)

The internal vibrations, which are practically the same
as for the free molecule. Some of their frequencies may
lie far above the phonon band(s) and are not likely to
be excited by phonon scattering. On the other hand, there
also might be low frequency modes below the maximum fre-
quency of the host lattice. Such modes usually are as-
sociated with the stretching motion involving heavy atoms
or bending modes. Bending vibrations have substantially
lower frequencies than stretching modes of the same bonds
(approximately 1/3 or even less [16, 3h, 35]). The reason
for this is'that bending motions primarily change angles
in the configuration of the participating points which do
not call for the same kind of restoring force (electro-
static repulsion) as in the case of stretching modes
where the bond length changes.

The translational vibrations of the whole molecule, which
are essentially the same as if the molecule was a single
mass. The dynamical behavior of point defects is quite
well understood [13, 32]. Also, the scattering problem
for this case has been treated already [13, 27,-29, 30,
36, 37] and we can take over the relevant results from
there.

The rotational vibrations (quaSi-rotations, librations)

of the whole molecule, for which the molecule acts as a

8h

rigid body with three moments of inertia. The coupling
of this type of motion to the host lattice will normally
be weak and the associated frequency is likely to be
found within the phonon band(s).

In the following Study we shall concentrate on the scattering of
phonons by molecular impurity centers. In many practical examples the fre-
quencies associated with motions of type (a) lie above the frequencies propa-
gated by the host crystal and will not affect the scattering cross section.
Furthermore, it would not be possible to set up a general model which accounts
for this type of motion and its coupling to the host lattice. Almost every
possible molecular defect (or at least each class of molecule) would require
a Special treatment and since we are more interested in possible general con-
clusions we Shall restrict our attention on the latter two types (types (b)
and (c)). There is, however, no justification for also neglecting the motion
of the center-of-mass of the defect molecule as was done by Wagner [25].
First, modes associated with this motion are most likely to be inband modes.
It is well known [13, 23, 38] that a heavier isotopic mass or the weakening
of the force constants around a point defect give rise to resonance (pseudo-
localized) modes. Then to be consistent with our model, where the center-
of-mass of the molecular defect belongs to the lattice system, and with the
assumption that the defect molecule be only weakly bound to the host crystal,

we have to expect that the force constants describing the links between the

molecular center-of-mass and the neighboring atoms are weaker than in the

ideal lattice. Second, as already mentioned in the discussion of the dif-

ferential cross section, Eq. (62), this mode might not only contribute

85

directly to the scattering cross section but also appreciably through the
interference term.

Let us now consider the rather simple model. As host lattice we
choose a monatomic (mass M) lattice of simple cubic structure with radial
force constantSIx and tangential force constants B. :The interaction among
the lattice points we restrict to nearest neighbors only. This crystal is

elastically stable as long as [39]

O<25<oz (67)

and the highest frequency prOpagated is

wiax : 21%.?21 (68)

We represent the molecular defect by a rigid sphere of a single moment of
inertia 6 and mass'M' which might be different from that of the atom re-
placed by the molecule. The sphere is coupled to the Six nearest lattice
atoms by tangential springs with constants f and radial springs with con-
stants k. Then the three remaining molecular coordinates are degenerate
and conveniently taken as the rotations ‘wx"wy’ we around the three cubic
axes. It is easy enough to see that this model allows motions of type (b)
as well as of type (c).

For this model the molecular Green's function is given by

 

7(m?) = -§-l--2 I (3 dimensional) (69)
w (x) -<n
where 0 hf
(D7114) = 2 (7O)

9/ a

86

There are 21 lattice coordinates involved in the disturbance,
namely, those to which the spherical molecule is coupled complemented by
the three coordinates of the center-of-mass. Our dynamical problem as well
as the matrix t are defined in this 21 dimensional defect space (matrix v).
However, group theory provides a powerful tool for reducing the calculation-
al effort, and we do not have to work in this high dimensional Space. From
the information given in Table I we see that the most that we must do is to
solve a 3 x 3 secular determinant for the modes transforming according to the

irreducible representation F It is also easy to see that only modes trans-

lu'
forming either according to the irreducible representation F18 or according

to F can induce dynamical effects in our model where we have replaced the

lu
molecular defect by a rigid Sphere. The former yield the librational motion
and the latter are connected with the motion of the center-of-mass. The
symmetry of the other modes is such that they provide no coupling which
could lead to a net force or torque.

If we denote by |s> the set of vectors which Span the defect space,

then the matrices a, b yfinz)‘b and g+ are defined'by their Hermite forms in

this Space:

I'th"

<slals> (k - 00 {52(Alg) + 32(Eé) + 52(E:) (

22 2 21 22 23
(f - B) {82(Flg) + s (F18) + 3 (F38) + 3 (F23) + s (F2g) + 3 (F23)

:Zhd

2
+ 32(Féu) + 82(Fgu) + s (Fin);

+ F(M,M',a,e,k,f) { 32(F1u) + 82(Fiu) + 82(Ffu)] (71)

87

 

2~ _f_ (1)2(K) 2 1 2 2 2
<S|b7(w )bls> ‘ M (D200 _ (DZ [8 (F12) + S (F12) + S (F728
<S|g+(dF)IS> = (3-6) ‘{SZ(A1g) + 32(E;) + 32(EZ) + s2(Fig) + 82(F7g) +

+ 82(Fgg) + 52(Fgg) I 82(Fgg)}
+ (i + e - 22) {82(Féu) + 82(Fgu) + 52(Fgu) ]
+ F(A,fi,e) {32(Fiu) + 32min) + 52(Ffu) }

where, assuming that we can define a longitudinal and two transversal

branches, the Green's function.3,B,e are given by

 

 

1 1 1
= = ___ Z + 22
) 1 + t 3N{Ea>2(bl) - ((02 + 16) EwZUSt) ' (“32 + 16) '

11>)
3»

Ag)

 

 

 

 

ia(k +k) ia(k +k)
B(‘1')=B1+Bezmfih2 2 +222 2
3 amen-(N +ie) mum-(III IIe)
A iZak iZak
C(w2)=e+8=—LZ e x +223 e x
1 t k—2 2 k 2 2
"w (151) - (w + is) ~a> (kt) - ((1) + is)

(72)

Zefg)

(73)

(7&8)

(7%)

(73¢)

The reason that the Hermite forms (71), (73) are determined only

Up to factors F(M,M',a,B,k,f), F(K,fi,e), respectively, is the following. As

we see, they are associated with that part of the defect space which is

88

spanned by vectors transforming according to the irreducible representation
Flu' This stable subspace is three dimensional, and solving the dynamical
problem (Lifshitz problem: Eq. (28a), B = O) we would obtain three equations
to determine the three free parameters (secular determinant). These results
then would enable us to give the factors above in explicit form. The solu-
tion of this problem involves Green's functions of the same type as 3,8,8,

respectively, and we know that they cannot be expressed analytically. A

good approximation is [23]

N 2 m = 2n
n sin 2 3 i i
G+(s) = (-1) fm (MN - 25171) c Jm1(271t) Jm2(272t) Jm3<2y3s at
0 ‘7 2 m1 = 2n + l
i
(75)
where‘g is the vector connecting the two points involved, the 71's denote
the force constants, and JK(x) is the Bessel function of order K. Evalu-
' 2
ation of Eq. (75) involves lengthy numerical computation for each [Bly<b ,
and combination of 05 B. We shall return to this problem later on.
With Eqs. (71), (72) and (73) we find the eigen values u(v) and
v(v) to be:
degeneracy
A ]_ 6
= A.- c - k - a (1) (7 8)
u<Alg) ( > M < )
" " 1 6b)
= - -' k " (I (2) (7
u(Eg) (A C) M ( )
A 1 <bzfg)
u(F = (A - c) a [f(1 - 2 2) - B] (3) (76c)
1g) (1) (x) - w
" 1 (76d)
1:: - C "" f " B) (3)
u(F2g) ( ) M (

89

 

degeneracy
“(F1u) = F(A:fi:6) F(M:M':Q’:B:k:f) (3) (769)
new) = (K + 6 - 21%) g (f - e) (3) was
1
v(Alg) = a (k - a) (1) ma)
veg) = ,l, (k - a) (2) (m)
1 Name)
v(Flg) == ,7 [f(1 - 2 2) - e] (3) (77¢)
ch (K) -<n
v(FZg) = 3 (f - a) (3) ma)
v(Flu) = F(M:MI:B:B:k) f) (3) (77E)
v(qu) =13} (f - e) (3) (77f)

where we have also indicated the degeneracy of the respective eigenvalue.
.Before using the results obtained so far to discuss the effect
of the individual modes on the scattering cross section we pause to con-
sider the effect of the requirement [to] that the potential energy be
invariant against infinitesimal rigid body rotations of the crystal. In

our case of a Simple cubic crystal we find that the condition imposed on

the tangential force constants is

f = B ; (78)

there is no restriction on the radial force constants. This has the con-

seQuence that the eigenvalues corresponding to the modes F28 and F2u van-

ish and we have to exclude them as possible eigenstates, since otherwise

90

this basic physical principle would be violated and the lattice would be-
come unstable. On the other hand, there are good reasons to relax this
condition. In a more realistic model we would also have to take into account
next nearest (and more distant) neighbors resulting in less severe restric-
tion of f, and, furthermore, we would also have to account for the change
in structure around the defect which might also affect these conditions.

In order to calculate the individual contributions to the scat-
tering cross section (Eq. (62)) we need, besides the ratios v(v)/(l + u(v)),
also matrix elements of the form <k'X'IT(v)|kK>. From the stable subspaces
given in Table I it is not difficult to construct the matrices T(v) (the
vector spanning the stable subspaces have to be normalized first) and the

nonvanishing matrix elements for each row of the different irreducible

representations are found to be:

<(k'OO)lIT(Alg)l(kOO) 1> <(k'OO) 1|T(A1g)l(0k0) 2> = <(k'OO) lIT(Alg)I(OOk) 3>

Sin (Ega) sin ka (79a)

H

mun)

<(Ok'O) 2IT(A1g)I(kOO) 1> <Ok'O) 2'T(Alg)|(OkO) 2> = <(Ok'O) 2|T(Alg)|(OOk) 3>

H

g Sin (gfa) sin ka (79b)

<(OOk') 3IT(A18)|(kOO) 1< <(OOk') 3|T(Alg)l(OkO) 2> = <(OOk') 3IT(A18)I(OOk) 3>

l
23- sin (gee) Sin ka (79c)

9

91

<(k'OO) llT(E;)l(kOO) 1> = <(k'OO) llT(E;)I(OkO) 2> = -.% <(k'OO) lIT(E;)|(OOk) 3>
= 5 sin (fie) sin ka ‘ (80a)

Duh—-

<(Ok'O) 2|T(E;)l(kOO) 1> = <(Ok'O) 2)T(E;)I(Ok0) 2> = - <(Ok'O) 2IT(E;)I(OOk)'3>

= 5 Sin (@a) Sin ka (80b)
<(OOk') 3'T(E;)l(kOO) 1> = <(OOk') 3|T(E;)I(Ok0) 2> = - % <(OOk') 3|T(E:)I(OOk) 3>
= - 5- sin (kga) sin ka (80C)

<(k'OO) 1IT(E:)](koo) 1> <(k'OO) llT(E:)I(OkO) 2>

= % sin (Ea) sin ka (81a)
((Ok'o) 2|T(E:)l(k00) 1> = - <(Ok'O) 2|T(E:)I(OkO) 2>
= - 8 Sin (fife) sin ka (81b)

<(OOk') 2|T(F}g)l(0ko) 3>

<(OOk') 2lT(F%g)I(OOk)2>

= - sin (fia) sin ka (828) .

<(0k'0) 3|T(F}g)l(00k)2> <(0k'o) 3IT(F{8)|(0ko) 3>

% Sin (Ea) sin ka (82b)

<(OOk')

<(k'OO)

<(Ok'O)

<(k'OO)

<(OOk')

<(Ok'0)

<(OOk')

<(k'OO)

2
1IT(F18)|(OOk)

2
3|T(Flg)|(00k)

lIT(F§g)I(OkO)

2|T(F§g)l(0ko)

2|T(F;g)|(00k)

H

H

H

H

>

>

>

>

3|T(F§g)|(00k) 2>

1|T(F:g)l(OOk)

2
3|T<Fzg)|(00k)

H

H

>

>

[I

92

. 2
<(OOk ) llT(Flg)|(kOO) 3>

sin (Ega) sin ka

ooh—-

I 2
<(k 00) 3|T(Flg)|(koo) 3>

% sin (Ega) sin ka

<(Ok'O) lIT(F§g)I(kOO) 2>

Sin (E23) sin ka

ULJIH

<(k'OO) 2|T(F§g)l(kOO) 2>

I
- 3 sin (kga) sin ka

<(OOk') 2IT(F;g)I(OkO) 3>

% sin (22a) sin ka

<(0k'o) 3|T(F;g)|(0ko) 3>

%~sin (gia) sin ka

<(OOk') 1|T(F:g)l(kOO) 3>

% sin (22a) sin ka

<(k'00) 3IT(F§g)|(k00) 3>

I I
% sin (kga) sin ka

(83a)

(83b)

(8ha)

(Bub)

(858)

(85b)

(86a)

(86b)

93

<(Ok'O) lIT(F3g)I(OkO) 1> <(Ok'O) 1|T(Fgg)|((koo) 2>

'% Sin (kya) sin ka

<(k'OO) 2|T(F3g)l(0ko) 1> <(k'OO) 2|T(F§g)|(koo) 2>

.% sin.(§ga) sin ka

<k'1IT(Fiu)I(kOO) 1>, 2g"1|T(Fiu)I(0ko) 1>, <k'llT(F1u)l(OOk) 1>
<5'2|T(F§u)|(koo) 2>, <k'2|T(FEu)I(OkO) 2>, <g'2IT(Ffu)|(00k) 2>
<t'3lT(Ffu)l(koo) 3>, <2'3|T(Ffu)|(0ko) 3>, <af3lT(F§u)l(00k) 3>

<(Ok'k) 1[T(F;u)|(0k0) 1>

<(Ok'k') lIT(F;u)[(OOk) 1>

_ [cos (£23) - cos ($28)] (cos ka - l)

<(k’Ok') 2[T(F:u)l(kOO) 2>

<(k'0k') 2|T(F§u)l(00k) 2>

[cos (Ega) - cos (figa)] (cos ka - l)

nut?

<(k'k'O) 3|T(Fgu)l(k00) 3>

<(k'k'o) 3|T(F§u)l(0ko) 3>

_ [cos (E28) - cos (Ega)] (cos ka - 1)

From these matrix elements we can learn a great deal about the

possible scattering processes

(878)

(an)

(88)

(89)

(90)

(91)

(92)

(93)

9h

This mode scatters longitudinally polarized phonons. It is

as"

acoustically active in the sense that the scattered acoustic
wave may be either longitudinally or transverse polarized.
E : This mode also scatters longitudinally polarized phonons

only and is acoustically active.

 

'11

This mode scatters transverse polarized phonons only and‘

 

the acoustical activity is restricted to transverse polarized
final states.
F2g : This mode also scatters transverse polarized phonons only
and has the same restricted acoustical activity as F18.
F1u : This mode scatters 32y incident phonon regardless of the
polarization but does not change the polarization.

qu : This mode scatters transverse polarized phonons only and

maintains the polarization.

From these results it is quite clear that, for example, the combin-
ation of modes which transform according to the irreducible representations
F18, Flu’ respectively, can give rise to a nonvanishing interference £232;
in Eq. (62).

We now ask under what conditions we might expect that one of the
modes contributes to the scattering cross section. One of the requirements,
of course, is that the associated frequency be inside the band(s) of the
ideal crystal. This information can be deduced from the results in the
detailed study of localized modes by Lengeler and Ludwig [39]. Under all

the conditions where they do not find a local mode-outside the ideal band,

there must be a resonance (pseudolocalized) mode inside the band. An

95

exception are the modes transforning according to the irreducible represen-
tation Flu where we have two modes either both localized, one localized and
one inband or both inside the ideal band. If one allows for a change in
mass only (isotope defect) then the solution of the dynamical problem is
rather simple. The 3 x 3 secular determinant reduces to just one equation

involving one Green's function g(m = 0) only:
2
(D €g(O)-1=O (9)4.)

where e = (M'-M) /M. For frequencies above the ideal band(s) (1)2 > (9205)..)
the Green's function above is negative definite and hence the condition to
find a localized mode is simply e < O or M' <TM. For frequencies within
the ideal band(s) the Green's function may be either positive or negative
definite and we may find solutions for e > O as well as for e < O. On the
other hand, we know that the problem has to have at least one solution and
since there is no solution outside the ideal band(s) for e > O we are bound
to find at least one solution inside the spectrum of the ideal lattice for
M' > M.

Let us consider this particular situation of a mass defect only
in more detail. Working within the acoustic approximation.Maradudin [36]

has derived the following expression for the total scattering cross section:

_ a6€2¢°br 1 ++) O<(D<(l) (958)
T 12II|ID(N2)|2 3(1) 0 (t) t

e a) 1 < < (95 )
2 -1r-- , an co (n1 b
12n[D(a> )I C (1)

a

 

96

where C(l), C(t) are the prOpagation velocities of the phonons in the lon-

an
1 d wt are

the corresponding Debye cutoff frequencies (ab) and Dflmz) is essentially

gitudinal branch and in the transversel branch, respectively,<n

the secular determinant of the dynamical problem.

We notice that in the long wavelength limit, i.e.,Im small, the
total cross section is proportional to w&. This result is the well known
Rayleigh scattering cross section. Of more interest, however, is the be-
havior of the cross section at somewhat higher frequencies. Here the fre-
quency dependence of the total cross section is determined largely by the
factor ID(co2) [-2.

Thoma and Ludwig [37] have plotted the term [Dhm2)I-2 for a number
of different values‘of s (Figure 2). They find a very strong resonance
peak for'M' = 6M in the region ofanmD = 0.2 and a smaller one for M' = 2M
atanuD = O.h5. To this result we shall return later on.

We turn now to the contribution due to the librational motion
of the defect molecule. For an incident phonon [(OOk) l> we find with

Eqs. (83a, b) the first term in Eq. (62) to be

1 , = a l t(F )
o ((OOk) 1, 1c.) 16“, 317;) I 18

 

Z O
I2 $5153 [engages + sinzces] <96)

and in the long wavelength limit the contribution to the total cross sec-

tion is

o;((OOk) 1) = 31-1; (RE-y)“ a2 (ka))+ |t(Flg)|2 (1) << ND (97)

h
We see that also in this case the Rayleigh scattering term (”k ) is modified,

 

97

 

’ I
6...
a /€'5
RN.“
.3
:5- §
5».
4- S
3
3..

 

 

 

 

 

 

 

 

 

Figure 2. [l - (1)2 e 3(0)]-2, which enters the scattering cross section
' of an isotope defect, as a function of the lattice wave frequen-
cy in the Debye approximation, taken from Thoma and Ludwig [37].

98

namely by the functional form of ]t(Flg)[2. The last expression can be

brought into the form

 

 

 

2
o; ((OOk) 1) = {—2} (-(-,%-t-5-)LL a2 (Du 33—“: BEBE-l- (98)
where C = 16 (fl/6)h/3‘DD. Wagner [25] has plotted the term It(Flg)I2/C
as a function of (wdwb)2 (Figure 3) and, assuming<mz(x) = 0.2 mg, found
a very strong resonance peak just below the molecular frequencyIm(x).
If we put the two figures (Figures 2, 3) on the same scales then

we note that the resonance peak for e = 1 (due to a heavy mass defect:
M' = 2M) occurs in the same region (w/cuD = 0.145 -—n. (cn/mD)2'->-’ 0.2) as
the molecular resonance of the librational mode and the ratio of the
peak heights is approximately 2.3/17.3. From this information we may
make a reasonable estimate about the magnitude of the interference term.
We assume that the two matrix elements (83) and (88) do not differ great-
ly and consider the product of the two factors t*(Flg) and t(F1u) only.
Its real part we express near the resonance as

Re -——L— - 1. = 51.71,. + 61 n, (99)

Gr 7 16i nr + lni (e: + e§)(n: + n?)
at the resonance we make the approximation
6
Re(t*(F1g) t(F1u)) 2 g n: (100)
E1 1)1

with this we get for the factor of the interference term

2 Re (t*(Flg) t(F1u)) a: 2 (2.3 . 17.3)“2 = 12.6 (101)

IO

99

 

 

 

 

 

 

 

 

l I 1 T I l l _17
.... III.I’/c _
L.
L.
_. (vi-012
1 l l_ 1 i L J l
.I .2 .3 .4 .5 .6 .7 .8 .9 to

Figure 3. It(F )[2, which enters the scattering cross section of the

libragional mode, as a function of the square modulus of the

lattice wave frequency in the Debye approximation, taken from
Wagner [25].

100

From this consideration we see that the interference term can be
of the same order as the larger of the two direct terms and its contribution

to the scattering cross section is certainly not negligible.

Ellipsoidal Molecules

We now replace in our model the rigid sphere by a rigid ellip-'
soid of which two moments of imentia are equal but different from that
with respect to the body C-axis. As we Shall be mostly interested in the
dynamical behavior of the librational modes in this case of an ellipsoidal
defect molecule, we assume the coupling to the lattice to be the same
as in the spherical case. Introducing this particular defect into the
host lattice results in a lower symmetry at the‘defect site depending
upon the orientation of the molecule with reapect to the crystallographic
axes. We shall consider the following three situations. The defect
molecule is oriented along one of the axes of the cube. In this case

the symmetry of the dynamical problem is Dh If the molecular defect is

h.
oriented along a body diagonal then we are dealing with the symmetry

group D The apprOpriate symmetry group for the molecule with its

3d°
C-axis parallel to one of the face diagonals is D2h'
We are primarily interested to see if there are associated
with the librational motion any new scattering mechanism (different
initial and final states) introduced by the non-Spherical defect mole-
cule which is conveniently done by looking at the elements of the scat-

tering matrix.

First we consider the situation where the ellipsoid is oriented

101

along a main axis. The necessary information to construct the correspond-
ing T(v) matrices is contained in Tables II and X. The matrix elements

different from zero are:

<(01éo) lIT(A2g)I(OkO) l> <(0k'o) 1IT(A28)I(kOO) 2>

= % Sin (E28) Sin ka (102a)

<(k'OO)'2[T(Azg)I(OkO 1> - <(k'OO) 2IT(A23)[(kOO) 2>

= - g-sin (Ega) sin ka (lO2b)

<(OOk') 1|T(E;)|(OOk) 1> <(OOk') 1|T(E;)I(kOO) 3>

= §~sin (figs) sin ka (103a)

<(k'OO) 3|T(E:)I(OOk) 1> <(k'OO) 3]T(E;)I(k00) 3>

= -'% sin (figs) sin ka (lO3b)

<(OOk') 2IT(E:)I(OOk) 2> <(OOk') 2IT(E:)I(OkO) 3>

=.% Sin (Ega) Sin ka I (loha)

<(Ok'O) 3IT(E:)I(OOk) 2> - <(Ok'O) 3IT(E:)I(OkO) 3>

% sin (figs) Sin ka (10kb)

102

We notice that, except for the different nomenclature and the fact that the
mode transforming according to the irreducible representation A2g is associa-
ted with the different moment of inertia, the matrix elements are exactly the
Same as in the spherical case (Eqs. (82), (83) and (8h)). However, the non-
degenerate mode interacts with phonons whose direction of incidence is perpen-
dicular to the defect axis only.

For the orientation along the body diagonal we find the stable

subspaces listed in Table III and the compatibility conditions are given

in Table XI. The following matrix elements are found to be different from

zero:

<(0k'k') lIT(AZg)I(0k0) 1> = -<(0k'k') lIT(Azg)I(00k) 1 >

- <(0k'k') lIT(AZg)I(k00) 2> s <(0k'k') 1IT(A28)I(00k) 2 >

<(0k'k') l[T(Azg)|(k00) 3> =-<(0k'k') 1[T(A28)|(0k0) 3 >

é-[Sin (Efa) -sin ($28)] sin ka (1053)

<(k'0k') 2|T(A23)I(0k0) 1> = —<(k'0k') 2|T(A28)|(00k) 1 >

- <(k'0k') 2IT(A28)I(k00) 2> <(k'0k') 2|T(A2g)l(00k) 2>

[I
ll

<(k'0k') 2|T(A28)I(k00) 3> -<(k'0k) 2[T(A28)[(Ok0) 3>

[sin (Ega) -sin (£29)] sin ka (thb)

\OIH

103

<(k'k'0) 3IT(A28)I(0kO) 1> = - <(k'k'0) 3IT(A28)I(00k) 1 >

- <(k'k'O) 3|T(A28)I(k00) 2> = <(k'k'0) 3|T(A2g)](00k) 2 >

<(k'k'o) 3|T(Azg)l(koo) 3> =-<(k'k'o) 3|T<A28)I(0k0) 3 >

= $- [sin (liga) -Sin (lfifafl sin ka (101m)
<(00k') lIT(E;)I(00k) 1> = <(OOk') l[T(E:)[(OOk) 2 >

= - <(OOk') lIT(E;)I(kOO) 3> = -<(00k') lIT(E:)[(0kO) 3 >

= % sin (k'ga) sin ka (105a)
<(00k') 2IT(E;)I(00k) 1> = <(00k') 2IT(E:)[(00k) 2 >

= - <(00k') 2[T(E;)[(k00) 3> = -<(00k') 2IT(E;)I(0k0) 3 >

s -16-sin (figs) sin ka (105b)
<(k'k'0) 3[T(E:)I(00k) 1> = <(k'k'0) 3IT(E;X(00k) 2 >

- <(k'k'0) 3IT(E;)I(koo) 3> = -<(k'k'o) 3|T(E;)l(0ko) 3 >

- 2E3“ (gigs) + sin (ggsflsin IN (105:)

10h

<(0k'k') 1|T(E:)|(0ko) 1> = 2<(0k'k') 1[T(E:)l(00k) 1 >

- <(0k'k') 1[T(E:)[(k00) 2> -2<(0k'k') 1IT(E:)[(00k) 2 >

-2<(0k'k') 1|T(E:)[(k00) 3> 2<(0k'k') lIT(E:)[(0k0) 3 >

éfsin (liga) + 2 sin (23.23)] Sin ka (106a)

<(k'0k') 2[T(E:)[(0k0) 1> = 2<(k'0k') 2[T(E:)[(00k) 1 >

- <(k'0k') 2[T(E:)I(k00) 2> -2<(k'0k') 2[T(E:)[(00k) 2 >

-2<(k'0k') 2[T(E:)[(k00) 3> 2<(k'0k') 2|T(E:)l(0ko) 3 >

= -%[Sin (152‘s) + 2 Sin (Egan sin ka (106b)
<(k'k'0) 3IT(E:)[(Ok0) 1> = 2<(k'k'0) 3[T(E:)[(00k) 1 >

-2<(k'k'0) 3[T(E:)[(00k) 2 >

- <(k'k'0) 3[T(E:)[(k00) 2>

-2<(k'k'0) 3[T(E:)[(k00) 3> 2<(k'k'0) 3|T(E:)[(0k0) 3 >

[sin (53a) - sin (E2a)] sin ka (106C)

\OIH

In this case we notice a considerable increase in the number of matrix elements,
but basically there is the same feature as for the spherical defect, namely,
that only transverse polarized phonons are scattered into transverse polarized

final states. It is also not difficult to see (e.g. by looking at the first

105

and second matrix element in Eq. (104a): <(0k'k') 1|T(A28)I(Ok0) 1>I=
-<(0k'k') l|T(A28)[(00k) l> and remembering that matrix elements of the
form <5}X'[T(A28)[(k00) L> are zero) that phonons incident parallel to the
axis of the defect are not scattered at all if their frequency corresponds
to the mode A28. In the limit when the ellipsoid degenerates into a Sphere,
of course, the corresponding matrix elements have to become equal. We dem-

onstrate this with the following example. We concentrate on the first

matrix elements in Eqs. (tha) and (1063)

<(0k'k') 1[T(A2g)[(0k0) l> = $[Sin (fife) - sin.(£§a)]sin ka

<(0k'k') l[T(E:)[(0k0) 1> =‘%[Sin (figs) + 2 sin (gfia)]sin ka

2
In the limiting case as mentioned above the two modes A2g and E8 become
degenerate and the two matrix elements appear in Eq. (62) (scattering cross

section) with the same factor. Therefore, we might add them to get the com-

bined contribution

sin (ggs) sin ka (107)

UHF‘

<(0k'k') 1[T(A2g) + T(E:)[(0k0) 1> =

This value corresponds to the first matrix element in Eq. (8ha),

<(0k'0) 1|T(FE ) [(OkO) l>, in agreement with the correlation table given
3
2
in Table XI according to which the mode Fig splits into A28 and E8 as the

symmetry is lowered from Oh to D3d'

For the molecule with its C-axis parallel to one of the face

diagonals of the cube we find all the.necessary information in Tables IV and

106

XII. The nonvanishing matrix elements are:

<(00k') 1[T(Blg)[(00k) 1> = - <(00k') l[T(Blg)[(00k) 2 >
=-<(00k') l|T(Blg)[(k00) 3> = <(00k') l[T(B18)[(0k0) 3 >

= % sin (@a) sin ka (1083)

<(00k') 2|T(Blg)[(00k) 1> = - <(00k') 2[T(Blg)[(00k) 2 >
=-<(00k') 2|T(Blg)[(k00) 3> = <(00k') 2[T(Blg)[(0k0) 3 >

= «2- Sin (k'é‘a) sin ka (108b)

<(k'k'0) 3[T(Blg)[(00k) 1>= - <(k'k'0) 3[T(Blg)[(00k) 2 >
’ =-<(k'k'0) 3|T(Blg)|(koo) 3> = <(k'k'o) 3IT(Blg)l(0k0) 3 >

= .% [sin (figs) - sin (k'gsn sin ka (108c)

<(00k') 1IT(B2g)I(00k) 1> = <(00k') 1[T(B28)[(00k) 2 >
=-<(00k') 1IT(22g)I(koo) 3 > = -<(00k') 1[T(B2g)[(0k0 3 >

= ésin (figs) sin ka 1 (1093)

107

<(00k') 2[T(B23)[(00k) 1> = <(00k') 2[T(B28)[(00k) 2 >

- <(00k') 2|T(Bzg)[(k00) 3> = - <(00k') 2[T(B28)[(Ok0) 3 >

'% sin (figs) sin ka (109b)

<(k'k'0) 3IT(B2g)[(00k) 1> = <(k'k'0) 3[T(B28)[(00k) 2 >

= ~ <(k'k'o) 3IT(BZg)l(k00) 3> = - <(k'k'0) 3]T(Bzg)l(0ko) 3 >

s -'%[Sin (figs) + sin (E2a)] sin ka (109s)
<(0k'0) 1[T(B3g)[(0k0) 1> s - <(0k'0) l[T(B3g)[(k00) 2 >

= I% Sin (gga) Sin ka (110a)
<(k'00) 2[T(B38)[(0k0) 1> = - <(k'00) 2[T(B3g)[(k00 2 >

s - é- sin (gas) sin ks (llOb)

As in the first two cases here the results also show that only transverse
polarized phonons are scattered by the librational motion and the acousti-
cal activity is restricted to transverse polarized final states. In our
calculations we assumed the ellipsoidal defect to be oriented along the

[llO]-direction. From the third and forth matrix element in Eq. (108a),

<(00k') l[T(Blg)[(k00) 3> = - <(OOk') 1[T(Blg)[(0k0) 3>

108

we see that phonons propagating parallel to the orientation of the molecule

are not affected by the librational mode B18 which is associated with the

different moment of inertia. In our model where we did not allow for any

changes in the force constants the modes B2g and B38 are degenerate.

VI. DISCUSSION
The group theoretical method presented in section II enabled us
to determine the stable subspaces which are spanned by eigenvectors cor-
responding to a certain eigenvalue for given symmetry operations. These
Stable subspaces are listed for the three basic cubic structures in the

case of full cubic symmetry and some of the subgroups of 0 We then went

h'
on to derive in section III compatibility relations for the components of
the stable subspaces of some of the subgroups. As first example, these
results were used in section IV to analyze the polarizations of the infra-
red active modes of a linear molecule imbedded in a cubic crystal, and it
was shown that in an ideal situation the direction of pOlarization, with
respect to the crystallographic axes, already determines uniquely the or-
ientation of the molecule within the crystal. As a second example we stud-
ied in section V the scattering of lattice waves by a stereoscopic defect
molecule in a simple cubic crystal in some detail. To do so we started

with a survey on a treatment most suitable to deal with this type of defects
as presented by wagner [2H, 25]. This Green's function technique enabled

us to remove the molecular coordinates and limit the defect space to the
same dimension as in the Lifshitz problem. The difference between this prob-
lem and the point defect problem is that in this case the effective dis-
turbance contains an additional term which has poles at the molecular
frequencies. In the neighborhood of these frequencies the effective distur-
bance cannot be treated as a perturbation. If one of the molecular frequen-
cies lies inside the ideal band(s) then we may find a resonance in the

scattering amplitude of lattice waves. In the next subsection we developed

109

 

110

a scattering formalism in terms of a T matrix and we were able to obtain a
formally exact solution of the scattering problem. We then derived an
expression for the differential scattering cross section which contained
two terms of equal importance. It was pointed out that there exists the
possibility that the interference term which had been neglected in the work
of Wagner may be of the same order of magnitude as the direct term. From
the form of the two terms in the scattering cross section it was realized
that the scattering of lattice waves by an impurity is much more complicated
than the scattering of plane waves by a static potential in quantum theory.
The equation which determines the stationary points can have solutions in
several branches of the functionIm2(kk) with the consequence that although
the incoming wave is in a definite branch ofIm2(kh), there can be several
scattered waves propagating in different directions with the same frequency
but with different group velocities and polarizations. From the expression
for the scattering cross section it was also seen that its resonances are
given by the resonances in the T matrix. We briefly discussed the condi-
tions for such resonances to occur and found that modes for which the
eigenvalues of the dynamical problem contain the poles of the molecular
Green's function are likely to satisfy the resonance condition. Assuming
that the internal binding in a molecule is much stronger.than the binding

to the host lattice, one can distinguish three types of motion for the
molecular defect: (a) the internal vibrations, (b) the translational vibra-
tions of the molecule as a whole, (c) the rotational vibrations of the whole
molecule. As it is not possible to describe motions of type (a) by a gen-

eral model we restricted our attention to motions of type'(b) and (c). We

 

111

pointed out that it would not be reasonable to also exclude motions of type

 

(b) eSpecially in view of the possibility that in combination with a libra-
tional mode they may give rise to a strong interference term in the scat-
tering cross section. First we considered the simple model of‘a rigid sphere
coupled to a simple cubic lattice with tangential as well as radial springs.
The stable subspaces which we determined at the beginning of this study
Simplified the solution of the eigenvalue problems to a large extent Since
we were able to define the relevant matrices by their Hermite forms in the

defect space. Imposing the condition resulting from the requirement that

 

the potential energy be invariant against infinitesimal rigid body rotations
of the crystal has the consequence that modes transforming according to the

irreducible representations F and FZu have to be excluded as possible

28
eigenstates,since they would lead to a local instability of the lattice.
Two reasons are given suggesting that we can relax this condition in a
more realistic situation. Then the matrix elements in the expression for
the differential cross section were calculated and from their particular
form we could draw the following conclusions:
1. Modes Alg and E8 interact with longitudinally polarized phonons
only and are acoustically active. 2. Modes Flg and F2g scatter transverse

polarized waves only and their acoustical activity is restricted to trans-

verse polarized final states. 3. The mode F scatters any incident

lu

phonon regardless of the polarization but does not change the polarization.

h. The mode F scatters transverse polarized phonons only without chang-
2u ’

ing their polarization. We then studied the conditions under which-we

expect that one or the other of the modes lies within the ideal band and

112

focused our attention on the situation of a mass defect motion of the sym-
metry Flu and the librational motion of a spherical molecule (symmetry F18).
In particular we considered the contribution to the scattering cross section.
In both instances we found a Rayleigh scattering term (vkh) modified in the
first case by a term depending upon the solution of the secular determinant
and in the latter case by the square modulus of the eigenvalue of the T
matrix corresponding to the mode Flg' We used the results obtained by
Thoma and Ludwig [37] and Wagner [25] to get a reasonable eatimate about
the magnitude of the interference term for the case where the resonances
due to the mode Flu and F18 occur at about the same frequency. We demon-
strated that under this circumstances the interference term is of the same
order as the larger of the two direct terms. In the next subsection we
replaced the rigid sphere by a rigid ellipsoid with two equal moments of
inertia, but different from the third one. We restricted our attention to
the librational modes only and also assumed the coupling to the lattice

to be the same as for the spherical molecule. Depending upon the orienta-
tion of the ellipsoid the symmetry at the defect Site is reduced either to

th (orientation along one of the principal axes of the cube), (orien-

D3d
tation along one of the body diagonals) or D2h (orientation parallel to

one of the face diagonals). With aid of the apprOpriate stable subspaces
and compatibility conditions we constructed the relevant matrix elements.
In all three cases we found basically the same matrix elements as for the
spherical defect molecule with the pr0perties that only transverse polarized

lattice waves are scattered and the acoustical activity is restricted to

transverse polarized final states. It was noted that the mode corresponding

 

 

 

113

to the different moment of inertia did not interact with phonons propagating
parallel to the orientation of the defect molecule.

We start the discussion of more realistic Situations with the
question, what happens if we let M}: M, k = Grand f = B, which, in case of
a point defect, corresponds to an ideal crystal. Therefore all the eigen-
values of the defect dynamical problem have to vanish since the transla-
tional symmetry of the lattice is no longer destroyed by a defect site and
acceptable solutions must have the proper point symmetry as well as trans-
lational symmetry. We note (Eqs. (76) and (77)) that all the eigenvalues
except those associated with the mode transforming according to the irre-

ducible representation F fulfill this requirement. The reason for this

Is
deviation is the following. The mode F1g corresponds to the librational
motion of the defect molecule, which is a consequence of the additional
degrees of freedom. The extended Green's function technique, although it
allowed us to exclude the molecular coordinates from our calculations, yet
the additional degrees of freedom must remain even when we change the para»
meters back to the ideal situation. We are thus dealing with a totally
different situation in the case of the "molecular defect" and we must then
exclude the mode Flg explicitly (Eqs. (76 c) and (77 1)).

Our expression for the scattering cross section (Eq. (62)) was
based on an acoustic approximation (we allowed, however, the propagation
velocity to be different in each branch). The correct form would have
contained second derivates of the surfaces of constant square modulus of the
frequency in‘krspace,-which we subsequently would have replaced by 2 c2(k)

anyway in order to use the results of Thoma and Ludwig [37] and wagner [25].

There are certainly limitations in the assumption of a Debye spectrum

 

11%

especially when Green's functions are involved. Calculating the Green's
functions at any given frequency we get contributions from the entire range
of the spectrum and the unrealistic singularity at the cutoff may reflect
itself in an unfavorable manner. However, the £232 of the matrix elements
is dependent on the symmetry of the defect problem alone, and a more real-
istic spectrum would affect the eigenvalues of the T matrix only. This means
that we would get exactly the same initial and final states but the reso-
nances might be shifted and their magnitudes altered.

Studying the librational motion of an elliptical molecule we
assumed that the force constants are the same as in the case of a spherical
defect. We now drop this assumption and ask if we could now couple to long-
itudinally polarized lattice waves under this circumstance. The necessary
but not sufficient condition is that in the decomposition of the modes
A18, Eg and F due to the lower symmetry (correlation table) there must

ls

be at least one irreducible representation in common to Flg and A.18 or Eg.

This is the case for the symmetries D (Table XI, Eg) and Dnh (Table XII,
C.

3d
33g). As mentioned above this condition is not sufficient and from the
compatibility conditions we see that the corresponding stable subspaces
are in fact mutually exclusive. There might be reasons to relax these com-
patibility conditions, for example, if the defect is no longer assumed to
be rigid. Then there exists the possibility that the neighboring atoms
might follow (energetic favorable) the internal motion of lower symmetry of
the defect, and are no longer governed by the over all cubic symmetry of
the crystal.

All calculations were performed within the harmonic approximation.

If the amplitude of any of the modes becomes too large to justify this

approximation all the obtained results become invalid as well.

 

 

 

 

LIST OF REFERENCE 8

REFERENCES
[1] R. F. Wallis, editor, Localized Excitations ifi.§2li§§ (Plenum
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[2] M. Wagner, Phys. Rev. 13%, lhh3 (1963).
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[5] J. A..A. Ketelaar et al., Spectrochim. Acta 13, 336 (1959).

 

[6] J. A. A. Ketelaar and C. J. H. Schutte, Spectrochim. Acta 11,

12:0 (1961).

 

[7] J. 1. Bryant and G. C. Turrell, J. Chem. Phys. 31, 1069 (1962).

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(John Wiley and Sons, Inc., New York, 194%).

[11] L. Jansen and M. Boon, Theory of Finite Groups, Applications in
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[12] E. Wigner, Gruppentheorie und ihre Anwendugg auf die Quantenmechanik
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[13] w. Ludwig,.Ergeb. Exakt. Naturw. 35, 1 (1963).

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